\begin{theorem}\label{Thm}
Let $\Ga$ be a connected tetravalent graph, let $(M, H)$ be a maximal $(\frac{1}{2}, t)$-pair of $\Ga$ with $t\geq1$, and let $K=\mathrm{Core}_H(M)$. Take $u\in V(\Ga)$. Then one of the followings holds.
\begin{enumerate}[\rm(a)]
\item\label{Thmcase:t=3} $t=3$, $(H/K, M/K, H_u, M_u)=(A_{72}, A_{71}, S_4\times S_3, \mz_2)$, and $\Ga$ is a cover of $\Ga_K$.
\item\label{Thmcase:t=2} $t=2$, $(H/K, M/K, H_u, M_u)= (S_5, \AGL_1(5), A_4, \mz_2)$, and $\Ga$ is a cover of $\Ga_K\cong \K_{5, 5}-5\K_2$.
\item\label{Thmcase:t=1} $t=1$, and either $M$ is normal in $H$, or $M_u^{\Ga(u)}\cong\mz_2$ is non-normal in $H_u^{\Ga(u)}\cong D_8$ such that $K$ is semiregular on $V(\Ga)$ with $|V(\Ga_K)|\neq2^\ell$ for any integer $\ell\geq0$ and one of the followings holds:
\begin{enumerate}[\rm (c1)]
\item\label{Thmcase:t=1_cover} $K$ is the kernel of $H$ acting on $V(\Ga_K)$, $\Ga$ is a cover of $\Ga_K$, and the pair $(M/K, H/K)$ is a maximal $(\frac{1}{2}, 1)$-pair of $\Ga_K$;
\item\label{Thmcase:t=1_cycle} $K$ is the kernel of $M$ acting on $V(\Ga_K)$, $\Ga_K\cong C_r$ with $r=|V(\Ga_K)|$, and $M/K\cong D_{2r}$.
\end{enumerate}
\end{enumerate}
Moreover, examples for each of the above cases exist.
\end{theorem}

All graphs considered in this paper are finite, undirected, regular and simple. Let $\Ga$ be a graph. For a positive integer $s$, a sequence $(u_0, u_1, \ldots, u_s)$ of vertices of $\Ga$ is said to be an \emph{$s$-arc} if $u_i$ is adjacent to $u_{i+1}$ for $0\leq i\leq s-1$ and $u_{j-1}\neq u_{j+1}$ for $1\leq j\leq s-1$. A $1$-arc is simply an \emph{arc}. We use $V(\Gamma)$, $E(\Gamma)$, $A(\Ga)$ and $\Aut(\Gamma)$ to denote its vertex set, edge set, arc set and (full) automorphism group, respectively.

Let $G$ be a subgroup of $\Aut(\Ga)$. If $G$ is transitive on $V(\Ga)$ and on $E(\Ga)$ but not on $A(\Ga)$, then we say that $\Ga$ is \emph{ $G$-half-arc-transitive}. In the literature, ``half-arc-transitive'' is also abbreviated as ``$\frac{1}{2}$-arc-transitive'' or ``HAT''. When $\Ga$ is $\Aut(\Ga)$-half-arc-transitive, we omit the prefix and simply say that $\Ga$ is half-arc-transitive (or HAT).
The study of HAT graphs was initiated by Tutte in \cite{Tutte1966}. Over the past half-century, substantial progress has been made, and we refer the reader to the survey articles \cite{CPS2015,Marusic1998} for an overview. Within this broader development, considerable attention has been devoted to tetravalent HAT graphs; see, for example, \cite{M-DM,MM1999,MN-JGT,MP1999,MX1997,PW-JCTB2007,RSparl2019,Spiga-Xia2021,Xia-2021,Zhou-JACO,Zhou-PAMS}.

For a positive integer $t$, we say that $\Ga$ is \emph{$(G,t)$-arc-transitive} if $G$ is transitive on the set of $t$-arcs of $\Ga$. If such a subgroup $G$ exists, then a $(G,t)$-arc-transitive graph will be simply called an \emph{$t$-arc-transitive graph}.
For $s\in\mathbb{N}\cup\{\frac{1}{2}\}$, if $\Ga$ is $(G,s)$-arc-transitive but not $(G,\lfloor s+1\rfloor)$-arc-transitive, then we say that $\Ga$ is \emph{$(G,s)$-transitive}. A pair $(M, H)$ of subgroups of $\Aut(\Ga)$ is called an \emph{$(s,t)$-pair} of $\Ga$ if $M\leq H\leq\Aut(\Ga)$ and $\Ga$ is $(M,s)$-transitive and $(H,t)$-transitive.
Such a pair $(M, H)$ is called \emph{maximal} if $M$ is a maximal subgroup of $H$.
Note that, $\Gamma$ admits a maximal $\bigl(\frac{1}{2},t\bigr)$-pair for some positive integer $t$ if and only if $\Gamma$ is $M$-half-arc-transitive and $H$-arc-transitive for some groups $M<H$ of automorphisms.
The following problem is both natural and of considerable interest.

\begin{problem}\label{prob}
Classify maximal $(s,t)$-pairs for all $s\in\mathbb{N}\cup\{\tfrac{1}{2}\}$ and $t\in\mathbb{N}$ with $s<t$.
\end{problem}

In this paper we address Problem~\ref{prob} for tetravalent graphs in the case $s=\tfrac{1}{2}$.
Our motivation is to better understand the automorphism groups of tetravalent $G$-HAT graphs $\Gamma$.
Knowing whether a given graph $\Gamma$ is HAT is often difficult, largely because determining $\Aut(\Gamma)$ is hard in general.
If $\Gamma$ is not half-arc-transitive, then $\Aut(\Gamma)$ is arc-transitive; consequently, $\Gamma$ admits a maximal $\bigl(\frac{1}{2},t\bigr)$-pair $(M,H)$ for some positive integer $t$ with $G\le M$.

Another motivation comes from studying the symmetry of the graph of alternating cycles associated with a $G$-HAT graph.
Let $\Ga$ be a tetravalent $M$-HAT graph for some $M\leq\Aut(\Ga)$. It is easy to see that the action of
$M$ on the arc set of $\Ga$ has two orbits, denoted by $O_M^+(\Ga)$ and $O_M^-(\Ga)$. For each $\epsilon\in \{+,-\}$,
and each edge of $\Ga$, the orbit $O_M^\epsilon(\Ga)$ contains exactly one of the two arcs corresponding to this edge.
Hence $O_M^+(\Ga)$ and $O_M^-(\Ga)$ give two opposite orientations of the edges of $\Ga$ and hence give two digraphs with the same vertex set as $\Ga$, denoted by $D_M^+(\Ga)$ and $D_M^-(\Ga)$.
A cycle of $\Ga$ is called an \emph{$M$-alternating cycle} if consecutive edges along the cycle have opposite orientations in $D_M^+(\Ga)$ (and hence also opposite orientations in $D_M^-(\Ga)$). In a seminal paper~\cite{Marusic1998} in 1998, Maru\v si\v c proved that all alternating cycles of the tetravalent $M$-HAT graph $\Ga$ have the same length, half of which is called the \emph{$M$-radius} of $\Ga$, and any two alternating cycles of $\Ga$ with nonempty intersection share the same number of vertices. This number is called the \emph{$M$-attachment number} of $\Ga$, denoted by $\att_M(\Ga)$.

The \emph{graph of $M$-alternating cycles} of $\Ga$, denoted $\Alt_M(\Ga)$, is the graph whose vertices are the $M$-alternating cycles of $\Ga$, with two such cycles adjacent whenever they have a non-empty intersection (see~\cite{MP1999,PSparl2017}). In~\cite{RSparl2019}, Rivera and \v{S}parl initiated the study of the automorphism group of $\Alt_M(\Ga)$ for tetravalent $M$-HAT graphs $\Ga$.
They proved that, if either $\att_M(\Ga)$ equals the $M$-radius or $\Aut(\Ga)$ contains an element interchanging $O_M^+(\Ga)$ and $O_M^-(\Ga)$, then $\Alt_M(\Ga)$ is arc-transitive (see~\cite[Proposition~5.7]{RSparl2019}). However, a tetravalent arc-transitive $M$-HAT graph need not admit an automorphism interchanging $O_M^+(\Ga)$ and $O_M^-(\Ga)$, and so this criterion does not always apply to assert the arc-transitivity of $\Alt_M(\Ga)$. Despite this, all computed examples in~\cite{RSparl2019} still have $\Alt_M(\Ga)$ arc-transitive. This led the authors to ask the following question for potential weaker conditions ensuring the arc-transitivity of $\Alt_M(\Ga)$, and similarly for the related quotient graph $\Ga_{\mathcal{B}(M)}$ (with respect to a certain $M$-invariant partition $\mathcal{B}(M)$ of $V(\Ga)$; see~\cite{MP1999} or~\cite{RSparl2019} for definition). In particular, $\Ga_{\mathcal{B}(M)}=\Ga$ whenever $\att_M(\Ga)=1$.

\begin{question}[{\cite[Question~5.8]{RSparl2019}}]\label{que:Sparl}
Is it true that, if $\Ga$ is a tetravalent arc-transitive graph which is $M$-HAT for some $M\leq\Aut(\Ga)$, the graphs $\Alt_M(\Ga)$ and $\Ga_{\mathcal{B}(M)}$ are both arc-transitive?
\end{question}

Note that any larger HAT group $N>M$ has the same orbits on $A(\Gamma)$ with $M$, hence induces the same graph of alternating cycles. We may let $M$ be maximal subject to that $\Ga$ is $M$-HAT. Take $H\leq\Aut(\Ga)$ such that $M$ is maximal in $H$. Then $H$ is arc-transitive on $\Ga$, and so $(M, H)$ is a maximal $(\frac{1}{2}, t)$-pair. If $M$ is normal in $H$, then $H$ preserves $\{O_M^+(\Ga),O_M^-(\Ga)\}$,  and so we deduce from~\cite[Proposition~5.7]{RSparl2019} that $\Alt_M(\Ga)$ is arc-transitive. Thus, the question is reduced to the case that $M$ is non-normal in $H$.

Our main theorem characterizes tetravalent graphs admitting a maximal $(\frac{1}{2}, t)$-pair with $t\geq1$. Recall that, for a subgroup $X$ of a group $Y$, the \emph{core} of $X$ in $Y$, denoted by $\mathrm{Core}_Y(X)$, is the largest normal subgroup of $Y$ contained in $X$.
Let $\Ga$ be a graph. Assume that $G\leq\Aut(\Ga)$ is such that $G$ is transitive on the vertex set of $\Ga$. Let $N$ be a normal subgroup of $G$ such that $N$ is intransitive on $V(\Ga)$. The \emph{normal quotient graph\/} $\Ga_N$ of $\Ga$ relative to $N$ is defined as the graph with vertices the orbits of $N$ on $V(\Ga)$ and with two different orbits adjacent if there exists an edge in $\Ga$ between the vertices lying in those two orbits. If $\Ga_N$ and $\Ga$ have the same valency, then $\Ga$ is a \emph{cover} of $\Ga_N$.

For a positive integer $t$, we say that $\Ga$ is \emph{$(G,t)$-arc-transitive} if $G$ is transitive on the set of $t$-arcs of $\Ga$. If such a subgroup $G$ exists, then a $(G,t)$-arc-transitive graph will be simply called an \emph{$t$-arc-transitive graph}.
For $s\in\mathbb{N}\cup\{\frac{1}{2}\}$, if $\Ga$ is $(G,s)$-arc-transitive but not $(G,\lfloor s+1\rfloor)$-arc-transitive, then we say that $\Ga$ is \emph{$(G,s)$-transitive}. A pair $(M, H)$ of subgroups of $\Aut(\Ga)$ is called an \emph{$(s,t)$-pair} of $\Ga$ if $M\leq H\leq\Aut(\Ga)$ and $\Ga$ is $(M,s)$-transitive and $(H,t)$-transitive.
Such a pair $(M, H)$ is called \emph{maximal} if $M$ is a maximal subgroup of $H$.
Note that, $\Gamma$ admits a maximal $\bigl(\frac{1}{2},t\bigr)$-pair for some positive integer $t$ if and only if $\Gamma$ is $M$-half-arc-transitive and $H$-arc-transitive for some groups $M<H$ of automorphisms.
The following problem is both natural and of considerable interest.

Another motivation comes from studying the symmetry of the graph of alternating cycles associated with a $G$-HAT graph.
Let $\Ga$ be a tetravalent $M$-HAT graph for some $M\leq\Aut(\Ga)$. It is easy to see that the action of
$M$ on the arc set of $\Ga$ has two orbits, denoted by $O_M^+(\Ga)$ and $O_M^-(\Ga)$. For each $\epsilon\in \{+,-\}$,
and each edge of $\Ga$, the orbit $O_M^\epsilon(\Ga)$ contains exactly one of the two arcs corresponding to this edge.
Hence $O_M^+(\Ga)$ and $O_M^-(\Ga)$ give two opposite orientations of the edges of $\Ga$ and hence give two digraphs with the same vertex set as $\Ga$, denoted by $D_M^+(\Ga)$ and $D_M^-(\Ga)$.
A cycle of $\Ga$ is called an \emph{$M$-alternating cycle} if consecutive edges along the cycle have opposite orientations in $D_M^+(\Ga)$ (and hence also opposite orientations in $D_M^-(\Ga)$). In a seminal paper~\cite{Marusic1998} in 1998, Maru\v si\v c proved that all alternating cycles of the tetravalent $M$-HAT graph $\Ga$ have the same length, half of which is called the \emph{$M$-radius} of $\Ga$, and any two alternating cycles of $\Ga$ with nonempty intersection share the same number of vertices. This number is called the \emph{$M$-attachment number} of $\Ga$, denoted by $\att_M(\Ga)$.

The \emph{graph of $M$-alternating cycles} of $\Ga$, denoted $\Alt_M(\Ga)$, is the graph whose vertices are the $M$-alternating cycles of $\Ga$, with two such cycles adjacent whenever they have a non-empty intersection (see~\cite{MP1999,PSparl2017}). In~\cite{RSparl2019}, Rivera and \v{S}parl initiated the study of the automorphism group of $\Alt_M(\Ga)$ for tetravalent $M$-HAT graphs $\Ga$.
They proved that, if either $\att_M(\Ga)$ equals the $M$-radius or $\Aut(\Ga)$ contains an element interchanging $O_M^+(\Ga)$ and $O_M^-(\Ga)$, then $\Alt_M(\Ga)$ is arc-transitive (see~\cite[Proposition~5.7]{RSparl2019}). However, a tetravalent arc-transitive $M$-HAT graph need not admit an automorphism interchanging $O_M^+(\Ga)$ and $O_M^-(\Ga)$, and so this criterion does not always apply to assert the arc-transitivity of $\Alt_M(\Ga)$. Despite this, all computed examples in~\cite{RSparl2019} still have $\Alt_M(\Ga)$ arc-transitive. This led the authors to ask the following question for potential weaker conditions ensuring the arc-transitivity of $\Alt_M(\Ga)$, and similarly for the related quotient graph $\Ga_{\mathcal{B}(M)}$ (with respect to a certain $M$-invariant partition $\mathcal{B}(M)$ of $V(\Ga)$; see~\cite{MP1999} or~\cite{RSparl2019} for definition). In particular, $\Ga_{\mathcal{B}(M)}=\Ga$ whenever $\att_M(\Ga)=1$.

\begin{question}[{\cite[Question~5.8]{RSparl2019}}]\label{que:Sparl}
Is it true that, if $\Ga$ is a tetravalent arc-transitive graph which is $M$-HAT for some $M\leq\Aut(\Ga)$, the graphs $\Alt_M(\Ga)$ and $\Ga_{\mathcal{B}(M)}$ are both arc-transitive?
\end{question}

Note that any larger HAT group $N>M$ has the same orbits on $A(\Gamma)$ with $M$, hence induces the same graph of alternating cycles. We may let $M$ be maximal subject to that $\Ga$ is $M$-HAT. Take $H\leq\Aut(\Ga)$ such that $M$ is maximal in $H$. Then $H$ is arc-transitive on $\Ga$, and so $(M, H)$ is a maximal $(\frac{1}{2}, t)$-pair. If $M$ is normal in $H$, then $H$ preserves $\{O_M^+(\Ga),O_M^-(\Ga)\}$, and so we deduce from~\cite[Proposition~5.7]{RSparl2019} that $\Alt_M(\Ga)$ is arc-transitive. Thus, the question is reduced to the case that $M$ is non-normal in $H$.

Our main theorem characterizes tetravalent graphs admitting a maximal $(\frac{1}{2}, t)$-pair with $t\geq1$. Recall that, for a subgroup $X$ of a group $Y$, the \emph{core} of $X$ in $Y$, denoted by $\mathrm{Core}_Y(X)$, is the largest normal subgroup of $Y$ contained in $X$.
Let $\Ga$ be a graph. Assume that $G\leq\Aut(\Ga)$ is such that $G$ is transitive on the vertex set of $\Ga$. Let $N$ be a normal subgroup of $G$ such that $N$ is intransitive on $V(\Ga)$. The \emph{normal quotient graph\/} $\Ga_N$ of $\Ga$ relative to $N$ is defined as the graph with vertices the orbits of $N$ on $V(\Ga)$ and with two different orbits adjacent if there exists an edge in $\Ga$ between the vertices lying in those two orbits. If $\Ga_N$ and $\Ga$ have the same valency, then $\Ga$ is a \emph{cover} of $\Ga_N$.

Our main theorem characterizes tetravalent graphs admitting a maximal $(\frac{1}{2}, t)$-pair with $t\geq1$. Recall that, for a subgroup $X$ of a group $Y$, the \emph{core} of $X$ in $Y$, denoted by $\mathrm{Core}_Y(X)$, is the largest normal subgroup of $Y$ contained in $X$.
Let $\Ga$ be a graph. Assume that $G\leq\Aut(\Ga)$ is such that $G$ is transitive on the vertex set of $\Ga$. Let $N$ be a normal subgroup of $G$ such that $N$ is intransitive on $V(\Ga)$. The \emph{normal quotient graph\/} $\Ga_N$ of $\Ga$ relative to $N$ is defined as the graph with vertices the orbits of $N$ on $V(\Ga)$ and with two different orbits adjacent if there exists an edge in $\Ga$ between the vertices lying in those two orbits. If $\Ga_N$ and $\Ga$ have the same valency, then $\Ga$ is a \emph{cover} of $\Ga_N$.

Summarized in Section~\ref{sec:sum}, the proof of Theorem~\ref{Thm} is developed in Sections~\ref{sec:proofofmain} and~\ref{sec:examples}. More precisely, we first prove in Section~\ref{sec:proofofmain} that one of~\eqref{Thmcase:t=3}--\eqref{Thmcase:t=1} must occur, and then in Section~\ref{sec:examples} we construct examples for each of the cases~\eqref{Thmcase:t=3},~\eqref{Thmcase:t=2},~\eqref{Thmcase:t=1_cover} and~\eqref{Thmcase:t=1_cycle}. The example for~\eqref{Thmcase:t=1_cover} yields the following corollary, which provides a negative answer to Question~\ref{que:Sparl} of Rivera and \v{S}parl~\cite{RSparl2019}.

\begin{lemma}[{\cite[Theorem~1.1]{AAMPS2016}}]\label{lem:4HAT}
Let $\Ga$ be a tetravalent $G$-HAT graph, and let $N\trianglelefteq G$ such that $N$ has at least three orbits on $V(\Ga)$. Then one of the following holds:
\begin{enumerate}[{\rm (a)}]
 \item $\Ga$ is a cover of $\Ga_N$, $N$ is semiregular on $V(\Ga)$ and is the kernel of $G$ acting on $V(\Ga_N)$, and $\Ga_N$ is $(G/N)$-HAT.
 \item $\Ga_N$ is an $r$-cycle with $r=|V(\Ga_N)|$, and the induced permutation group of $G$ on $V(\Ga_N)$ is $\mz_r$ or $D_{2r}$.
\end{enumerate}
\end{lemma}

\begin{lemma}\label{lem:hat-2at}
Let $\Ga$ be a tetravalent graph, let $u\in V(\Ga)$, let $(M, H)$ be a maximal $(\frac{1}{2}, t)$-pair of $\Ga$ with $t\geq2$, and let $K=\mathrm{Core}_H(M)$. Then $\Ga$ is a cover of $\Ga_K$, and one of the following holds:
\begin{enumerate}[\rm (a)]
\item $t=2$, $(H/K, M/K, H_{u}, M_{u})=(S_5, \AGL_1(5), A_4, \mz_2)$, and $\Gamma_K\cong \K_{5, 5}-5\K_2$;
\item $t=3$, and $(H/K, M/K, H_{u}, M_{u})=(A_{72}, A_{71},S_4\times S_3, \mz_2)$.
\end{enumerate}
\end{lemma}

\begin{lemma}\label{lem:stab:=2}
Let $\Ga$ be a tetravalent graph, let $u\in V(\Ga)$, let $(M, H)$ be a maximal $(\frac{1}{2},1)$-pair of $\Ga$ such that $M_u^{\Ga(u)}\cong\mz_2$ and $H_u^{\Ga(u)}\cong D_8$, and let $K=\mathrm{Core}_H(M)$. Then $M_u^{\Ga(u)}\ntrianglelefteq H_u^{\Ga(u)}$, $K$ is semiregular on $V(\Ga)$ with $|V(\Ga_K)|\neq2^\ell$ for any integer $\ell\geq0$, and one of the following holds:
\begin{enumerate}[\rm (a)]
\item\label{lem:stab:=2-a} $K$ is the kernel of $H$ acting on $V(\Ga_K)$, the pair $(M/K, H/K)$ is a maximal $(\frac{1}{2}, 1)$-pair of $\Ga_K$, and $\Ga$ is a cover of $\Ga_K$;
\item\label{lem:stab:=2-b} $\Ga_K\cong C_r$ with $r=|V(\Ga_K)|$, $M/K\cong D_{2r}$, and $K$ is the kernel of $M$ acting on $V(\Ga_K)$.
\end{enumerate}
\end{lemma}

\begin{example}\label{exam:hat_in_3-trans_non-normal}
Let $M=\langle a,b\mid a^2=b^3=(ab)^2=1\rangle\times\langle c,d\mid c^2=d^3=(cd)^4=1\rangle\cong S_3\times S_4$, let $R$ be the faithful action of $M$ by right multiplication on $V:=[M\,{:}\,\langle a(cd)^2\rangle]$, let $X=\mathrm{Alt}(V)\cong A_{72}$, and let $Y=R(M'\rtimes\langle ac\rangle)\cong \mz_3\rtimes S_4$. Then $Y$ is a regular subgroup of $X$. Let $Z=R((\langle b\rangle\times\langle d\rangle)\rtimes\langle ac^{dc}\rangle)\cong \mz_3^2\rtimes \mz_2$. Computer searching shows that there exists an element $x$ of order $4$ in $\N_X(Z)\cap\mathbf{C}_X(ac^{dc})$ satisfying the following conditions:
\begin{itemize}
\item $x^2=ac^{dc}\in Z$;
\item $Y\cap Y^x=Z$;
\item $X=\langle Y,x\rangle$;
\item $YxY=YS$ for some $S=\{g,g^{-1},g^h,(g^h)^{-1}\}$, where $g\in X_v$ for some $v\in V$ and $h$ is an involution in $X_v$. \end{itemize}

Corollaries~\ref{cor-1} and~\ref{cor-2} follow from Examples~\ref{exam:hat_in_1-trans_altgra} and~\ref{exam:hat_in_3-trans_non-normal}, respectively. Now we prove Theorem~\ref{Thm}.
Let $\Ga$ be a connected tetravalent graph, let $(M, H)$ be a maximal $(\frac{1}{2}, t)$-pair of $\Ga$ with $t\geq1$, let $K=\mathrm{Core}_H(M)$, and let $u\in V(\Ga)$.
If $t\geq2$, then Lemma~\ref{lem:hat-2at} asserts that~\eqref{Thmcase:t=3} or~\eqref{Thmcase:t=2} holds.
Assume that $t=1$. If $M_u^{\Ga(u)}\cong\mz_2$ and $H_u^{\Ga(u)}\cong D_8$, then by Lemma~\ref{lem:stab:=2}, $M_u^{\Ga(u)}\cong\mz_2$ is non-normal in $H_u^{\Ga(u)}\cong D_8$ such that $K$ is semiregular on $V(\Ga)$ with $|V(\Ga_K)|\neq2^\ell$ for any integer $\ell\geq0$ and we have~\eqref{Thmcase:t=1_cover} or~\eqref{Thmcase:t=1_cycle}. If either $M_u^{\Ga(u)}\ncong\mz_2$ or $H_u^{\Ga(u)}\ncong D_8$, then since $(M, H)$ is a $(\frac{1}{2}, 1)$-pair of $\Ga$, it follows that $M_u^{\Ga(u)}$ is normal in
$H_u^{\Ga(u)}$ with $|H_u^{\Ga(u)}:M_u^{\Ga(u)}|=2$. In this case, Lemma~\ref{lem:hat-1t}\,\eqref{lem:hat-1t-c} implies that $M$ is normal in $H$, as in case~\eqref{Thmcase:t=1} of Theorem~\ref{Thm}. Moreover, according to Examples~\ref{exam:hat_in_1-trans_altgra},~\ref{exam:hat_in_3-trans_non-normal},~\ref{exam:hat_in_2-trans} and~\ref{exam:hat_in_1-trans_cycle}, examples for each of the cases~\eqref{Thmcase:t=3},~\eqref{Thmcase:t=2},~\eqref{Thmcase:t=1_cover} and~\eqref{Thmcase:t=1_cycle} of Theorem~\ref{Thm} exist. This completes the proof.

Summarized in Section~\ref{sec:sum}, the proof of Theorem~\ref{Thm} is developed in Sections~\ref{sec:proofofmain} and~\ref{sec:examples}. More precisely, we first prove in Section~\ref{sec:proofofmain} that one of~\eqref{Thmcase:t=3}--\eqref{Thmcase:t=1} must occur, and then in Section~\ref{sec:examples} we construct examples for each of the cases~\eqref{Thmcase:t=3},~\eqref{Thmcase:t=2},~\eqref{Thmcase:t=1_cover} and~\eqref{Thmcase:t=1_cycle}. The example for~\eqref{Thmcase:t=1_cover} yields the following corollary, which provides a negative answer to Question~\ref{que:Sparl} of Rivera and \v{S}parl~\cite{RSparl2019}.

\begin{corollary}\label{cor-1}
There exists a connected tetravalent graph $\Ga$ admiting a maximal $(\frac{1}{2}, 1)$-pair $(M,\Aut(\Gamma))$ such that $\Alt_M(\Ga)$ is half-arc-transitive.
\end{corollary}

Another byproduct of proving Theorem~\ref{Thm} is the discovery of the first example of a tetravalent normal-edge-transitive but non-normal Cayley graph on a nonabelian simple group.
Given a group $G$ and an inverse-closed generating subset $S\subseteq G\setminus\{1\}$, the \emph{Cayley graph} $\Cay(G,S)$ on $G$ is the graph with vertex set $G$ and edge set $\{\{g, sg\} \mid g\in G, s\in S\}$. Denote by $R_G$ the right multiplication action of $G$ on itself, and denote
\[
\Aut(G, S)=\{\a\in\Aut(G)\mid S^\a=S\}.
\]
It is well known~\cite{Godsil1981} that the normalizer of $R_G(G)$ in $\Aut(\Cay(G,S))$ is $R_G(G)\rtimes\Aut(G,S)$. We call $\Cay(G,S)$ \emph{normal-edge-transitive} if this normalizer is transitive on $E(\Cay(G,S))$, and \emph{normal} if this normalizer equals $\Aut(\Cay(G,S))$.
In 1999, Praeger~\cite{Praeger1999} proved that every connected normal-edge-transitive Cayley graph of valency $3$ on a nonabelian simple group is normal. This result was recently extended to all prime valencies in~\cite{Zhang-F-Z-Y}, where it was also shown that there exist connected normal-edge-transitive non-normal Cayley graphs of valency $8$ on nonabelian simple groups. However, it has remained open whether such graphs exist at smaller valencies, particularly valency $4$, the smallest possible valency, despite substantial work on tetravalent edge-transitive Cayley graphs on nonabelian simple groups~\cite{FangLiXu,FangWZ}. Our example for Theorem~\ref{Thm}\,\eqref{Thmcase:t=3} resolves this question by establishing the following:

\begin{corollary}\label{cor-2}
There exists a connected tetravalent normal-edge-transitive non-normal Cayley graph on $A_{71}$.
\end{corollary}

The post-theorem introduction says that “the proof of Theorem~\ref{Thm} is developed in Sections~\ref{sec:proofofmain} and~\ref{sec:examples}.” More precisely: (1) “we first prove in Section~\ref{sec:proofofmain} that one of~\eqref{Thmcase:t=3}--\eqref{Thmcase:t=1} must occur,” and then (2) “in Section~\ref{sec:examples} we construct examples for each of the cases~\eqref{Thmcase:t=3},~\eqref{Thmcase:t=2},~\eqref{Thmcase:t=1_cover} and~\eqref{Thmcase:t=1_cycle}.”

The post-theorem introduction says that “the proof of the main theorem is developed later.” More precisely: (1) “we first prove later that one of the following must occur: \begin{enumerate}[\rm(a)]
\item\label{Thmcase:t=3} $t=3$, $(H/K, M/K, H_u, M_u)=(A_{72}, A_{71}, S_4\times S_3, \mz_2)$, and $\Ga$ is a cover of $\Ga_K$.
\item\label{Thmcase:t=2} $t=2$, $(H/K, M/K, H_u, M_u)= (S_5, \AGL_1(5), A_4, \mz_2)$, and $\Ga$ is a cover of $\Ga_K\cong \K_{5, 5}-5\K_2$.
\item\label{Thmcase:t=1} $t=1$, and either $M$ is normal in $H$, or $M_u^{\Ga(u)}\cong\mz_2$ is non-normal in $H_u^{\Ga(u)}\cong D_8$ such that $K$ is semiregular on $V(\Ga)$ with $|V(\Ga_K)|\neq2^\ell$ for any integer $\ell\geq0$ and one of the followings holds:
\begin{enumerate}[\rm (c1)]
\item\label{Thmcase:t=1_cover} $K$ is the kernel of $H$ acting on $V(\Ga_K)$, $\Ga$ is a cover of $\Ga_K$, and the pair $(M/K, H/K)$ is a maximal $(\frac{1}{2}, 1)$-pair of $\Ga_K$;
\item\label{Thmcase:t=1_cycle} $K$ is the kernel of $M$ acting on $V(\Ga_K)$, $\Ga_K\cong C_r$ with $r=|V(\Ga_K)|$, and $M/K\cong D_{2r}$.
\end{enumerate}\end{enumerate},” and then (2) “later we construct examples for each of the cases given by the items \ref{Thmcase:t=3} and \ref{Thmcase:t=2} above, and by the subcases \begin{enumerate}[\rm (c1)]
\item\label{Thmcase:t=1_cover} $K$ is the kernel of $H$ acting on $V(\Ga_K)$, $\Ga$ is a cover of $\Ga_K$, and the pair $(M/K, H/K)$ is a maximal $(\frac{1}{2}, 1)$-pair of $\Ga_K$;
\item\label{Thmcase:t=1_cycle} $K$ is the kernel of $M$ acting on $V(\Ga_K)$, $\Ga_K\cong C_r$ with $r=|V(\Ga_K)|$, and $M/K\cong D_{2r}$.
\end{enumerate}.”

\label{Thm}
Let $\Ga$ be a connected tetravalent graph, let $(M, H)$ be a maximal $(\frac{1}{2}, t)$-pair of $\Ga$ with $t\geq1$, and let $K=\mathrm{Core}_H(M)$. Take $u\in V(\Ga)$. Then one of the followings holds.
\begin{enumerate}[\rm(a)]
\item\label{Thmcase:t=3} $t=3$, $(H/K, M/K, H_u, M_u)=(A_{72}, A_{71}, S_4\times S_3, \mz_2)$, and $\Ga$ is a cover of $\Ga_K$.
\item\label{Thmcase:t=2} $t=2$, $(H/K, M/K, H_u, M_u)= (S_5, \AGL_1(5), A_4, \mz_2)$, and $\Ga$ is a cover of $\Ga_K\cong \K_{5, 5}-5\K_2$.
\item\label{Thmcase:t=1} $t=1$, and either $M$ is normal in $H$, or $M_u^{\Ga(u)}\cong\mz_2$ is non-normal in $H_u^{\Ga(u)}\cong D_8$ such that $K$ is semiregular on $V(\Ga)$ with $|V(\Ga_K)|\neq2^\ell$ for any integer $\ell\geq0$ and one of the followings holds:
\begin{enumerate}[\rm (c1)]
\item\label{Thmcase:t=1_cover} $K$ is the kernel of $H$ acting on $V(\Ga_K)$, $\Ga$ is a cover of $\Ga_K$, and the pair $(M/K, H/K)$ is a maximal $(\frac{1}{2}, 1)$-pair of $\Ga_K$;
\item\label{Thmcase:t=1_cycle} $K$ is the kernel of $M$ acting on $V(\Ga_K)$, $\Ga_K\cong C_r$ with $r=|V(\Ga_K)|$, and $M/K\cong D_{2r}$.
\end{enumerate}
\end{enumerate}
Moreover, examples for each of the above cases exist.,

Question: Let \(\Gamma\) be a connected tetravalent graph. Suppose \(M\le H\le \operatorname{Aut}(\Gamma)\), \(M\) is a maximal subgroup of \(H\), \(\Gamma\) is \(M\)-half-arc-transitive (that is, \(M\) is transitive on vertices and edges but not on arcs), and \(\Gamma\) is \((H,t)\)-transitive for some integer \(t\ge 1\); equivalently, \((M,H)\) is a maximal \((\tfrac12,t)\)-pair. Let \(K=\operatorname{Core}_H(M)\), the largest normal subgroup of \(H\) contained in \(M\), and fix \(u\in V(\Gamma)\). Write \(\Gamma(u)\) for the neighbor set of \(u\), \(H_u\) and \(M_u\) for the stabilizers of \(u\), and \(G_u^{\Gamma(u)}\) for the permutation group induced by \(G_u\) on \(\Gamma(u)\). Since \(K\trianglelefteq H\), let \(\Gamma_K\) be the normal quotient graph whose vertices are the \(K\)-orbits on \(V(\Gamma)\), with two \(K\)-orbits adjacent whenever some edge of \(\Gamma\) joins them. A subgroup is semiregular on \(V(\Gamma)\) if no nonidentity element fixes a vertex, and saying that \(\Gamma\) is a cover of \(\Gamma_K\) means the natural quotient map is a covering projection. Which statement holds for every such \(\Gamma,M,H,K,u\)?

One of the following holds. (a) \(t=3\), \((H/K,\,M/K,\,H_u,\,M_u)=(A_{72},\,A_{71},\,S_4\times S_3,\,\mathbb Z_2)\), and \(\Gamma\) is a cover of \(\Gamma_K\). (b) \(t=2\), \((H/K,\,M/K,\,H_u,\,M_u)=(S_5,\,\operatorname{AGL}_1(5),\,A_4,\,\mathbb Z_2)\), and \(\Gamma\) is a cover of \(\Gamma_K\cong K_{5,5}-5K_2\) (the graph obtained from \(K_{5,5}\) by deleting a perfect matching). (c) \(t=1\), and either \(M\trianglelefteq H\), or else \(M_u^{\Gamma(u)}\cong\mathbb Z_2\) is non-normal in \(H_u^{\Gamma(u)}\cong D_8\), \(K\) is semiregular on \(V(\Gamma)\), \(|V(\Gamma_K)|\neq 2^\ell\) for every integer \(\ell\ge 0\), and in addition either (c1) \(K\) is the kernel of the action of \(H\) on \(V(\Gamma_K)\), \(\Gamma\) is a cover of \(\Gamma_K\), and \((M/K,H/K)\) is a maximal \((\tfrac12,1)\)-pair of \(\Gamma_K\); or (c2) \(K\) is the kernel of the action of \(M\) on \(V(\Gamma_K)\), \(\Gamma_K\cong C_r\) with \(r=|V(\Gamma_K)|\), and \(M/K\) is dihedral of order \(2r\). Moreover, examples realizing each listed case exist.

Choices:
B. One of the following holds. (a) \(t=3\), \((H/K,\,M/K,\,H_u,\,M_u)=(A_{72},\,A_{71},\,S_4\times S_3,\,\mathbb Z_2)\), and \(\Gamma\) is a cover of \(\Gamma_K\). (b) \(t=2\), \((H/K,\,M/K,\,H_u,\,M_u)=(S_5,\,\operatorname{AGL}_1(5),\,A_4,\,\mathbb Z_2)\), and \(\Gamma\) is a cover of \(\Gamma_K\cong K_{5,5}-5K_2\). (c) \(t=1\), and either \(M\trianglelefteq H\), or else \(M_u^{\Gamma(u)}\cong\mathbb Z_2\) is non-normal in \(H_u^{\Gamma(u)}\cong D_8\), \(K\) is semiregular on \(V(\Gamma)\), \(|V(\Gamma_K)|\neq 2^\ell\) for every integer \(\ell\ge 1\), and in addition either (c1) \(K\) is the kernel of the action of \(H\) on \(V(\Gamma_K)\), \(\Gamma\) is a cover of \(\Gamma_K\), and \((M/K,H/K)\) is a maximal \((\tfrac12,1)\)-pair of \(\Gamma_K\); or (c2) \(K\) is the kernel of the action of \(M\) on \(V(\Gamma_K)\), \(\Gamma_K\cong C_r\) with \(r=|V(\Gamma_K)|\), and \(M/K\) is dihedral of order \(2r\). Moreover, examples realizing each listed case exist.
C. One of the following holds. (a) \(t=3\), \((H/K,\,M/K,\,H_u,\,M_u)=(A_{72},\,A_{71},\,S_4\times S_3,\,\mathbb Z_2)\), and \(\Gamma\) is a cover of \(\Gamma_K\). (b) \(t=2\), \((H/K,\,M/K,\,H_u,\,M_u)=(S_5,\,\operatorname{AGL}_1(5),\,A_4,\,\mathbb Z_2)\), and \(\Gamma\) is a cover of \(\Gamma_K\cong K_{5,5}-5K_2\). (c) \(t=1\), and either \(M\trianglelefteq H\), or else \(M_u^{\Gamma(u)}\cong\mathbb Z_2\) is non-normal in \(H_u^{\Gamma(u)}\cong D_8\), \(K\) is semiregular on \(V(\Gamma)\), and in addition either (c1) \(K\) is the kernel of the action of \(H\) on \(V(\Gamma_K)\), \(\Gamma\) is a cover of \(\Gamma_K\), and \((M/K,H/K)\) is a maximal \((\tfrac12,1)\)-pair of \(\Gamma_K\); or (c2) \(K\) is the kernel of the action of \(M\) on \(V(\Gamma_K)\), \(\Gamma_K\cong C_r\) with \(r=|V(\Gamma_K)|\), and \(M/K\) is dihedral of order \(2r\).
D. One of the following holds. (a) \(t=3\), \((H/K,\,M/K,\,H_u,\,M_u)=(A_{72},\,A_{71},\,S_4\times S_3,\,\mathbb Z_2)\), and \(\Gamma_K\) is a cover of \(\Gamma\). (b) \(t=2\), \((H/K,\,M/K,\,H_u,\,M_u)=(S_5,\,\operatorname{AGL}_1(5),\,A_4,\,\mathbb Z_2)\), and \(\Gamma_K\cong K_{5,5}-5K_2\) with \(\Gamma_K\) a cover of \(\Gamma\). (c) \(t=1\), and either \(M\trianglelefteq H\), or else \(M_u^{\Gamma(u)}\cong\mathbb Z_2\) is non-normal in \(H_u^{\Gamma(u)}\cong D_8\), \(K\) is semiregular on \(V(\Gamma)\), \(|V(\Gamma_K)|\neq 2^\ell\) for every integer \(\ell\ge 0\), and in addition either (c1) \(K\) is the kernel of the action of \(H\) on \(V(\Gamma_K)\), \(\Gamma_K\) is a cover of \(\Gamma\), and \((M/K,H/K)\) is a maximal \((\tfrac12,1)\)-pair of \(\Gamma_K\); or (c2) \(K\) is the kernel of the action of \(M\) on \(V(\Gamma_K)\), \(\Gamma_K\cong C_r\) with \(r=|V(\Gamma_K)|\), and \(M/K\) is dihedral of order \(2r\). Moreover, examples realizing each listed case exist.
E. One of the following holds. (a) \(t=3\), \((H/K,\,M/K,\,H_u,\,M_u)=(A_{72},\,A_{71},\,S_4\times S_3,\,\mathbb Z_2)\), and \(\Gamma\) is a cover of \(\Gamma_K\). (b) \(t=2\), \((H/K,\,M/K,\,H_u,\,M_u)=(S_5,\,\operatorname{AGL}_1(5),\,A_4,\,\mathbb Z_2)\), and \(\Gamma\) is a cover of \(\Gamma_K\cong K_{5,5}-5K_2\). (c) \(t=1\), and either \(M\trianglelefteq H\), or else \(M_u^{\Gamma(u)}\cong\mathbb Z_2\) is normal in \(H_u^{\Gamma(u)}\cong D_8\), \(K\) is semiregular on \(V(\Gamma)\), \(|V(\Gamma_K)|\neq 2^\ell\) for every integer \(\ell\ge 0\), and in addition either (c1) \(K\) is the kernel of the action of \(H\) on \(V(\Gamma_K)\), \(\Gamma\) is a cover of \(\Gamma_K\), and \((M/K,H/K)\) is a maximal \((\tfrac12,1)\)-pair of \(\Gamma_K\); or (c2) \(K\) is the kernel of the action of \(M\) on \(V(\Gamma_K)\), \(\Gamma_K\cong C_r\) with \(r=|V(\Gamma_K)|\), and \(H/K\) is dihedral of order \(2r\). Moreover, examples realizing each listed case exist.

dropped condition |V(\Gamma_K)|\neq 2^\ell for all \ell\ge 0

direction of covering projection to normal quotient

non-normal local action and quotient group in cycle case


\begin{theorem}\label{t:main}
	Let $G$ be an $n$-vertex digraph with $\alpha^*(G) \le s$.
	Then, $f(G) \ge n^{5/9-o(1)}s^{-14/9}$. In particular, if $\alpha^*(G)=n^{o(1)}$, then
	$f(G) \ge n^{5/9-o(1)}$.
\end{theorem}

Throughout this paper, the chromatic number $\chi(G)$, the independence number $\alpha(G)$, and the bipartite independence number $\alpha^*(G)$\,\footnote{Recall that $\alpha(G)$ is the largest $k$ for which there is an empty $k$-vertex set and that $\alpha^*(G)$ is the largest $k$ for which there are two disjoint $k$-vertex sets with no edges between them.} of a digraph $G$ are inherited from the corresponding parameters of its underlying undirected graph.
An {\em orientation} (a.k.a. oriented graph) is a digraph without directed cycles of length $2$. A {\em tournament} is an orientation of a complete graph. A digraph is {\em acyclic} if it has no directed cycles.

Graph colorings and orientations are closely related topics whose origins date back to the
classical theorem of Gallai–Hasse–Roy–Vitaver \cite{gallai-1968,hasse-1965,roy-1967,vitaver-1962}
asserting that every orientation of a $k$-chromatic graph has a directed path with $k$ vertices.
Many variations and related results connecting these concepts can be found in the survey \cite{havet-2013}.
A very general variant of the Gallai–Hasse–Roy–Vitaver Theorem was conjectured by Burr \cite{burr-1980},
stating that any $(2k-2)$-chromatic digraph contains every oriented tree on $k$ vertices.
The special case of Burr's conjecture for tournaments is known as
Sumner's conjecture which has been resolved, for $k$ sufficiently large, by K\"uhn, Mycroft, and Osthus
\cite{KMO-2011}. The best bound on the chromatic number of digraphs which guarantees the containment of every oriented tree on $k$ vertices
is $O(k^{1.5})$ which was recently obtained by Bessy, Gon{\c{c}}alves, and Reinald \cite{BGR-2024}.

Burr's conjecture is known to hold for acyclic digraphs. In fact, it was proved by
Addario-Berry, Havet, Sales, Reed, and Thomass\'e \cite{addario+-2013} that every $k$-chromatic acyclic digraph
contains every oriented tree on $k$ vertices. It would therefore be of interest to prove that
digraphs have acyclic subgraphs that retain the original chromatic number to some nontrivial extent.
Indeed, this problem was formulated in \cite{addario+-2013}.
For a digraph $G$, let $f(G)$ denote the maximum chromatic number of an acyclic subgraph of $G$
and let $f(k)$ be the minimum of $f(G)$ taken over all $k$-chromatic digraphs.
It is a folklore argument (see below) that $f(G) \ge \sqrt{\chi(G)}$ (whence $f(k) \ge \sqrt{k}$), and it was proved in \cite{addario+-2013}
that $f(k) \ge 1+\lfloor \sqrt{k-1} \rfloor$. Trivially, $f(3)=2$ and it is known that $f(4)=3$ \cite{NY-2019,shapira-2022}. The special case of this parameter for the case of tournaments was considered
by Nassar and the author in \cite{NY-2019}. Let $g(k)$ be the maximum integer such that every $k$-vertex tournament has an acyclic subgraph with chromatic number $g(k)$. It was conjectured in \cite{NY-2019}
that $g(k) = \omega(\sqrt{k})$, namely that the folklore lower bound can be improved by more than a constant factor. This conjecture has recently been resolved by Fox, Kwan, and Sudakov \cite{FKS-2021} who proved by an
ingenious argument that $g(k)=\Omega(k^{5/9-o(1)})$. In that same paper, they have shown that the exponent $5/9$
cannot be improved beyond $3/4$. Returning back to $f(k)$, this result suggests that $f(k)=\omega(\sqrt{k})$ as well.
Our main result shows that this is the case in a dense regime.

We shall require the following result of Gallai and Milgram \cite{GM-1960} (see also \cite{diestel-2005}).
\begin{lemma}\label{l:gm}{\rm\cite{GM-1960}}
 Let $G$ be a digraph with $\alpha(G)=s$. Then there is a set of $s$ pairwise vertex-disjoint directed paths of $G$ that cover all its vertices.
\end{lemma}

For the rest of this paper, all digraphs are assumed to be orientations. Clearly this assumption does not change the chromatic number, independence number, or bipartite independence number.
For a subset $S$ of vertices of an oriented graph $G$, let $G[S]$ denote the induced subgraph of $G$ on $S$.
We say that $S$ is an {\em acyclic $k$-set} if $|S|=k$ and $G[S]$ is acyclic.
\begin{lemma}\label{l:prob}
 Let $G$ be a $k$-vertex orientation with $\alpha(G) \le s$. For a random permutation $\pi$, the probability that $G_\pi$ has no edges is at most $(se/k)^k$.
\end{lemma}
\begin{proof}
 By Lemma \ref{l:gm}, $G$ has a set of $s$ pairwise vertex-disjoint paths $P_1,\ldots,P_s$ that cover each vertex.
 Let $p_i$ denote the number of vertices of $P_i$ (possibly $p_i=0$) and observe that $\sum_{i=1}^s p_i = k$.
 Notice that for $G_\pi$ to have no edges, all vertices of $P_i$ must
 appear in the reverse order they appear in $P_i$, which, for a random $\pi$, occurs with probability
 $1/p_i!$. As the $P_i$'s are pairwise vertex-disjoint, we have that the probability that
 $G_\pi$ has no edges is therefore at most the reciprocal of $\prod_{i=1}^s p_i!$.
 Finally note that by logarithmic convexity of $x^x$,
 $$
 \prod_{i=1}^s p_i! \ge \prod_{i=1}^s \left( \frac{p_i}{e}\right)^{p_i} \ge \left( \frac{\sum_{i=1}^s p_i}{se}\right)^{\sum_{i=1}^s p_i}=\left( \frac{k}{se}\right)^{k}\;.
 $$
\end{proof}
\begin{corollary}\label{coro:prob}
 If $G$ is an $n$-vertex orientation with $\alpha(G) \le s$ and $G$ has fewer than $(k/se)^k$ acyclic $k$-sets, then there is a permutation $\pi$ such that $\alpha(G_\pi) \le k$. In particular, $\chi(G_\pi) \ge n/k$.
\end{corollary}
\begin{proof}
 Consider a random $\pi$. For a set $S$ of $k$ vertices, the
 probability that $S$ is an independent set in $G_\pi$ is either $0$ if $S$ is not acyclic, or, by Lemma \ref{l:prob}, at
 most $(se/k)^k$ if $S$ is acyclic. Hence, if $G$ has fewer than $(k/se)^k$ acyclic $k$-sets, then by the union bound, $\alpha(G_\pi) \le k$ with positive probability.
\end{proof}

\begin{lemma}\label{l:deg}
 Let $G$ be an $m$-vertex orientation with $\alpha(G) \le s$. Then $G$ has a vertex whose in-degree and out-degree are both at least $m/4s-1/2$.
\end{lemma}
\begin{proof}
 Let $t=m/4s-1/2$. Assume the claim is false and partition $V(G)$ into parts $V_1$ and $V_2$ where all vertices of $V_1$ have out-degree smaller than $t$ and all vertices of $V_2$ have in-degree smaller than $t$. Without loss of generality, $|V_1| \ge m/2$. Consider the underlying undirected graph of $G[V_1]$. Since $\alpha(G[V_1]) \le s$, there is a subgraph of this undirected graph
 with minimum degree at least $|V_1|/s-1 \ge m/2s-1$. Returning to the directed version, there must be some vertex in this subgraph whose out-degree is not smaller than its in-degree, hence its out-degree is at least $(m/2s-1)/2=t$, contradicting the definition of $V_1$.
\end{proof}

\begin{lemma}\label{l:l22}
 Let $k = n^{\Omega(1)}$ and $k \ge s^7$. If an $n$-vertex orientation $G$ with $\alpha(G) < s$ and $\alpha^*(G) < s$ has at least $(k/se)^k$ acyclic $k$-sets, then it has a set of $n' \ge k^{2-o(1)}/s$ vertices inducing a $q$-almost-acyclic subgraph $G'$
 where $q = n's^6/k^{1-o(1)}$.
\end{lemma}

\begin{lemma}\label{l:l23}
Suppose $n^{1/3}s^{-2/3} \le q \le n^{1-\Omega(1)}$. An $n$-vertex $q$-almost-acyclic orientation $G$ with
$\alpha(G) < s$ has a permutation $\pi$ with $\alpha(G_\pi) \le n^{1/4+o(1)}q^{1/4}s^{3/2}$, so $\chi(G_\pi) \ge n^{3/4-o(1)}q^{-1/4}s^{-3/2}$.
\end{lemma}

\begin{lemma}\label{l:l24}
 Let $n$ be sufficiently large and let $G$ be an $n$-vertex orientation with $\alpha(G) < s$ and $\alpha^*(G) < s$ having at least
 $M$ acyclic $k$-sets. If $q = 6000s^6 n\log^2n/k$, then there is $k'$ satisfying $k-3k/\log n \le k' \le k$ such that $G$ has an induced $(q,6sk/\log n)$-almost-acyclic subgraph $G'$ on at least $e^{-8}kM^{1/k}$ vertices having at least $e^{-6k}M$ acyclic $k'$-sets.
\end{lemma}
\begin{proof}
 Let $t$ be a positive integer parameter to be set later.
 For each $i=0,\ldots,t$ we define vertex sets $W_i,V_i,V'_i$ such that
 \begin{enumerate}
 \item
 $W_i$ is an ordered acyclic $i$-set, $W_i \subset W_{i+1}$, and the ordering of $W_{i+1}$ is consistent with the ordering of $W_i$.
 \item
 $V_0'=V(G)$, $V_i' \supseteq V_i \supset V_{i+1}' \supseteq V_{i+1}$ for $i=0,\ldots,t-1$.
 \item
 $V_i' \cap W_i = \emptyset$.
 \end{enumerate}
 Describing the iterative procedure building these sets requires several notions.
 \begin{itemize}
 \item
 Let $H'_i$ be the $(k-i)$-uniform hypergraph with vertex set $V'_i$ whose edges are the
 $(k-i)$-subsets $S$ for which $G[S \cup W_i]$
 is an acyclic $k$-set for which there is a topological sort $\sigma_S$ consistent with the ordering of $W_i$.
 Let $N'_i$ denote the number of edges of $H'_i$ and observe that $N'_0 \ge M$.
 \item
 Let $H_i$ be an induced subhypergraph of $H'_i$ with minimum degree at least $N'_i/|V'_i|$ (it will always be the case that $N'_i > 0$). Such a subhypergraph can be obtained by repeatedly removing vertices with degree smaller than
 $N'_i/|V'_i|$ until none are left. Let $N_i$ be the number of edges of $H_i$
 and let $V_i$ be its vertex set.
 \item
 Let the ordering of $W_i$ be $\{w_1,\ldots,w_i\}$ so that if $(w_j,w_{j'}) \in E(G)$, then $j < j'$.
 Consider some vertex $v \in V$ and some edge $S$ of $H_i$ containing $v$.
 Recall that $G[S \cup W_i]$ is an acyclic $k$-set for which $\sigma_S$ is a topological sort consistent with the ordering of $W_i$. So if $(w_j,v) \in E(G)$ and $(v, w_{j'}) \in E(G)$, it must be that $j < j'$. Thus, all out-neighbors of $v$ in $W_i$ (if there are any) appear in $W_i$'s ordering after all in-neighbors of $v$ in $W_i$ (if there are any). We say that the set of {\em allowed bins} of $v$ with respect to $W_i$ is $\{j,\ldots,j'\}$
 if $(w_j,v) \in E(G)$, $(v,w_{j'+1}) \in E(G)$ and $v$ is not adjacent in $G$ to any of
 $w_{j+1},\ldots,w_{j'}$. If $v$ has no in-neighbors in $W_i$, then let $j=0$ and if $v$ has no out-neighbors in $W_i$, then let $j'=i$. Notice that for any edge $S$ of $H_i$ containing $v$,
 $\sigma_S$ places $v$ after some $w_\ell$ and before some $w_{\ell+1}$ where $j \le \ell \le j'$ (if $\ell = 0$, then $j=0$ so $v$ is placed before $w_1$ and if $\ell =i$, then $j'=i$ so $v$ is placed after all $w_i$). We say that bin $\ell$ is {\em associated with $(\sigma_S,v)$}. One may think of $\{w_1,\ldots,w_i\}$ as labeling the consecutive internal walls of a set of $i+1$ bins labeled $0,\ldots,i$.
 \item
 Two vertices $x,y$ of $V_i$ are {\em incompatible} if $(x,y) \in E(G)$ and every allowed bin of $x$ with respect to $W_i$ is {\em after} every allowed bin of $y$ with respect to $W_i$
 (in particular, no edge of $H_i$ contains both $x$ and $y$).
 \end{itemize}
 For $0 \le i \le t-1$, we shall go from step $i$ to step $i+1$ by picking some (carefully selected) vertex $w \in V_i$ and some (carefully selected) allowed bin $\ell$ of $w$. We add $v$ to $W_i$ to obtain $W_{i+1}$ with the ordering of $W_{i+1}$ being $\{w_1,\ldots,w_\ell,w,w_{\ell+1},\ldots,w_i\}$.
 We remove from $V_i$ all vertices incompatible with $w$ to obtain $V'_{i+1}$. The edges of $H'_{i+1}$ are obtained by taking all edges $S$ of $H_i$ which contain $w$ and for which $(\sigma_S,w)$ is associated with $\ell$, and keeping $S \setminus w$ as an edge of $H'_{i+1}$.

\noindent
Theorem \ref{t:main} extends the result of Fox, Kwan, and Sudakov which is the case $s=0$.
Notice that if $\alpha^*(G) \le s$, then $\alpha(G) \le 2s+1$ and thus $\chi(G) \ge n/(2s+1)$.
So, Theorem \ref{t:main} significantly improves upon the folklore $\sqrt{\chi(G)}$ lower bound
for graphs with relatively small bipartite independence number.
Theorem \ref{t:main} is particularly interesting in the setting of {\em random graphs}.
Let ${\cal G}(n,p)$ denote the  Erd\H{o}s–R\'enyi random $n$-vertex graph with edge probability $p$.
It is well known that if $p = n^{-o(1)}$, then $G \sim {\cal G}(n,p)$ almost surely\footnote{As usual, ``almost surely'' means with probability tending to $1$ as $n$ tends to infinity.} has
$\chi(G) =n^{1-o(1)}$ and $\alpha^*(G)=n^{o(1)}$ (see \cite{bollobas-1988} for sharp estimates when $p$ is constant). We therefore have:
\begin{corollary}\label{coro:main}
	Let $p = n^{-o(1)}$ and $G \sim {\cal G}(n,p)$. Then almost surely,
	every orientation $\vec{G}$ of $G$ has $f(\vec{G}) \ge n^{5/9-o(1)}$. Using $p=1/2$ we have that almost all $n$-vertex graphs $G$ have the property that every orientation $\vec{G}$ of $G$ has $f(\vec{G}) \ge n^{5/9-o(1)}$.
\end{corollary}

To prove Theorem \ref{t:main}, we first use a theorem of Gallai and Milgram on the path-covering number of digraphs with given independence number. This is then used, together with some additional ideas, to effectively
reduce the proof to the same framework of the proof of Fox, Kwan, and Sudakov. We do, however, need to (sometimes extensively) modify their arguments in locations where their proof relies on the fact that every pair of vertices is adjacent, which is not the case in our setting. In Section \ref{sec:lem} we prove several lemmas needed for our proof.
In Section \ref{sec:fks} we revisit the proof of Fox, Kwan and Sudakov, state variants of their
Lemmas 2.2 (which is Lemma \ref{l:l22} here) and 2.3 (which is Lemma \ref{l:l23} here) and prove
Lemma \ref{l:l23}. Lemma \ref{l:l22}, which is the main technical lemma, is proved in Section \ref{sec:l22}. Throughout the paper, we ignore rounding issues as long as they do not affect the asymptotic claims.

We end the introduction recalling the aforementioned folklore lower bound. Clearly, in every red/blue edge coloring
of a graph $G$, $\chi(G_r) \cdot \chi(G_b) \ge \chi(G)$ where $G_r$ and $G_b$ respectively denote the red and blue subgraphs. For every digraph $G$ and every linear order of its vertices, we can color red the edges going from left to right and color blue the edges going from right to left, observing that the obtained $G_r$ and $G_b$ are acyclic. Hence $\max(\chi(G_r),\chi(G_b)) \ge \sqrt{\chi(G)}$.
Nevertheless, despite the similarity between acyclic orientations and red/blue edge colorings, note that an analogous result such as Theorem \ref{t:main} is, in general, false for red/blue edge colorings. To see this, consider the complete $n/s$-partite
graph with each part of order $s$; clearly in this case $\alpha^*(G) \le \alpha(G)=s$.
Further partition the parts into $\sqrt{n/s}$ buckets, each bucket consisting of$\sqrt{n/s}$ parts. Now color blue all the edges with endpoints in parts belonging to the same bucket
and color red all remaining edges. It is immediate to verify that $G_r$ and $G_b$ are both $\sqrt{n/s}$ chromatic.

To prove the main theorem, we first use a theorem of Gallai and Milgram on the path-covering number of digraphs with given independence number. This is then used, together with some additional ideas, to effectively reduce the proof to the same framework of the proof of Fox, Kwan, and Sudakov. We do, however, need to (sometimes extensively) modify their arguments in locations where their proof relies on the fact that every pair of vertices is adjacent, which is not the case in our setting. Next we prove several lemmas needed for our proof. We then revisit the proof of Fox, Kwan and Sudakov, state variants of their Lemmas 2.2 (which is
\begin{lemma}\label{l:l22}
	Let $k = n^{\Omega(1)}$ and $k \ge s^7$. If an $n$-vertex orientation $G$ with $\alpha(G) < s$ and $\alpha^*(G) < s$ has at least $(k/se)^k$ acyclic $k$-sets, then it has a set of $n' \ge k^{2-o(1)}/s$ vertices inducing a $q$-almost-acyclic subgraph $G'$
	where $q =  n's^6/k^{1-o(1)}$.
\end{lemma}
here) and 2.3 (which is
\begin{lemma}\label{l:l23}
Suppose $n^{1/3}s^{-2/3} \le q \le n^{1-\Omega(1)}$. An $n$-vertex $q$-almost-acyclic orientation $G$ with
$\alpha(G) < s$ has a permutation $\pi$ with $\alpha(G_\pi) \le n^{1/4+o(1)}q^{1/4}s^{3/2}$, so $\chi(G_\pi) \ge n^{3/4-o(1)}q^{-1/4}s^{-3/2}$.
\end{lemma}
here) and prove the lemma above. The lemma above, which is the main technical lemma, is proved later.

Question: Let $G$ be an $n$-vertex digraph. Define $\alpha^*(G)$, the bipartite independence number of $G$, to be the largest integer $k$ for which there exist two disjoint $k$-vertex sets with no edges between them in the underlying undirected graph of $G$. Define $f(G)$ to be the maximum chromatic number of an acyclic subgraph of $G$, where acyclic means having no directed cycle. Assume that $\alpha^*(G) \le s$. Which conclusion about $f(G)$ holds?

One has $f(G) \ge n^{5/9-o(1)} s^{-14/9}$. In particular, if $\alpha^*(G)=n^{o(1)}$, then $f(G) \ge n^{5/9-o(1)}$.

Choices:
B. One has $f(G) \ge n^{3/5-o(1)} s^{-3/2}$. In particular, if $\alpha^*(G)=n^{o(1)}$, then $f(G) \ge n^{3/5-o(1)}$.
C. If $\alpha^*(G)=n^{o(1)}$, then $f(G) \ge n^{5/9-o(1)}$.
D. There is an absolute constant $c>0$ such that for every $n$-vertex digraph $G$ with $\alpha^*(G) \le s$, one has $f(G) \ge c\,n^{5/9}s^{-14/9}$.
E. One has $f(G) \ge n^{5/9-o(1)} s^{-14/9}$ provided both $\alpha(G) \le s$ and $\alpha^*(G) \le s$. In particular, if $\alpha(G)=\alpha^*(G)=n^{o(1)}$, then $f(G) \ge n^{5/9-o(1)}$.

combined-exponent from q-almost-acyclic/FKS reduction

dropped the general quantitative bound in terms of s, retaining only the stated special case

non-uniform o(1) losses replaced by a uniform absolute constant

extra hypothesis alpha(G) <= s imported from intermediate lemmas


\begin{cor}
\label{cor:cycles}
The graph $\Gkp$ contains cycles
of lengths $4n+2$, $6n+3$, and $8n+2$ if there exists $n\in\mathbb{Z}_{>0}$ such that $\Knkpprop\ne\emptyset$.
\end{cor}

\begin{cor}
\label{cor:sufficient condition non-planarity}
The graph $\Gkp$ is non-planar if there exists $n\in\mathbb{Z}_{>0}$ such that $\Knkpprop\ne\emptyset$.
\qed
\end{cor}

\begin{theorem}
\label{thm:main}
Let $\kappa \in\mathbb{Z}\setminus \{4\}$.
Then the followings hold:
\begin{enumerate}
\item[$(1)$]
$(\mathrm{Theorems}~\ref{thm-n=1},\ \ref{thm:n=2 non-planarity} \  \text{and}\ \ref{thm-n=(p-1)/2})$
For each such $\kappa$ and for infinitely many primes $p$ (depending on $\kappa$) of natural density at least $1/2$,
there exists an explicit $K_{3,3}$-subdivision in $\Gkp$.

\item[$(2)$]
$(\mathrm{Theorem}~\ref{thm:np density})$
Moreover, for almost all $\kappa$,
the natural density of primes $p$ for which $(1)$ holds is at least $13/16$.

\end{enumerate}
\end{theorem}

\begin{theorem}
\label{thm-proj}
Let $p>3$ be a prime. Then we have the followings.
\begin{enumerate}
\item[$(1)$] The graph $\Gzp$ is toroidal if and only if $p=7$.
\item[$(2)$] The graph $\Gzp$ is projective-planar if and only if $p=7$.
\end{enumerate}
\end{theorem}

The Diophantine equation
\begin{equation}
\label{def:Markoff-equation}
x^2+y^2+z^2=xyz
\end{equation}
is called the {\it Markoff equation},
and a solution $(x,y,z)$ is called a {\it Markoff triple}.
For a prime $p>3$,
let $\Mzp$ denote the set of all Markoff triples over $\Fp$, the finite field with $p$ elements.
The {\it Markoff mod $p$ graph} $\Gzp$ is a finite $3$-regular graph (possibly with loops) whose vertex set is $\Mzp \setminus \{(0,0,0)\}$, and two triples are adjacent if and only if they are transformed into each other by one of the following {\it Vieta involutions}:
\[
R_1(x,y,z)=(yz-x,y,z), \quad
R_2(x,y,z)=(x,zx-y,z), \quad
R_3(x,y,z)=(x,y,xy-z).
\]

The graph $\Gzp$ has been extensively studied in the literature on ``strong approximation'' for the Markoff equation (\ref{def:Markoff-equation}).
Indeed, Bourgain, Gamburd, and Sarnak~\cite{BGS2016} conjectured that $\Gzp$ is connected for all primes $p>3$.
Recently Chen~\cite{C2024} made a breakthrough toward this conjecture by proving that $\Gzp$ is connected for any sufficiently large prime $p$.
A key result here is that the size of any component of $\Gzp$ must be divisible by $p$. Furthermore, for any $\varepsilon>0$, there are only $O(p^{\varepsilon})$ vertices that are not contained in the giant component (\cite{BGS2016}) while $\Gzp$ has approximately $p^2$ vertices (see (\ref{eq-number}) below).
Martin~\cite{Martin2025} provided an elementary proof of Chen's result concerning the divisibility of the component sizes of $\Gzp$.
In addition, Eddy, Fuchs, Litman, Martin, and Tripeny~\cite{EFLMT2025} established an explicit lower bound for primes $p$ for which the connectivity of $\Gzp$ is guaranteed, namely, $p>3.448\times10^{392}$.
There are several algorithmic results towards the connectivity of $\Gzp$, and it was verified in \cite{B2025} that $\Gzp$ is connected for $p<10^6$.

On the other hand, de Courcy-Ireland~\cite{C2024} established the non-planarity of $\Gzp$: $\Gzp$ is planar if and only if $p=7$. This provides (indirect but positive) evidence for another well-known conjecture by Bourgain, Gamburd, and Sarnak~\cite{BGS2016, BGS2026} asserting that $\{\Gzp\}_{\text{$p$:prime}}$ is in fact an expander family, which would imply higher connectivity for $\Gzp$ (for further details on expander family, see e.g. \cite{HLW2004, L2012}).
The proof of non-planarity consists of two parts: an asymptotic evaluation of the Euler characteristic
of $\Gzp$ and explicit constructions of subdivisions of the complete bipartite graph $K_{3,3}$.
The Kuratowski-Wagner theorem shows a complete characterization for planar graphs in terms of graph minors, that is, a graph is planar if and only if it contains neither $K_{3,3}$ nor the complete graph $K_5$ as a graph minor.
Since $\Gzp$ is a $3$-regular graph, $K_{3,3}$ is the unique obstruction to planarity of $\Gzp$ in the sense that
a $3$-regular graph cannot contain a subdivision of $K_5$, and more generally, any $3$-regular graph containing $K_5$ as a minor must also contain a subdivision of $K_{3,3}$.

There are several generalizations of \eqref{def:Markoff-equation}, such as the following equation: for a (rational) integer $\kappa$,
\begin{equation}
\label{def:Markoff-type equation}
x^2+y^2+z^2=xyz+\kappa.
\end{equation}
Throughout this paper, we naturally treat $\kappa$ as an element of $\Fp$ when considering the solutions of \eqref{def:Markoff-type equation} modulo $p$.
For the set of all such solutions over $\Fp$, denoted by $\Mkp$,
Carlitz \cite{C1954} proved that
\begin{align}
\label{eq-number}
 \# \Mkp
=p^2+\left(3+\left(\frac{\kappa}{p}\right)\right)\left(\frac{\kappa-4}{p}\right)p+1,
\end{align}
where $(\frac{\cdot}{p})$ is the Legendre symbol.
The {\it generalized Markoff mod $p$ graph} $\Gkp$ (with respect to \eqref{def:Markoff-type equation}) is defined as a $3$-regular graph
whose vertex set is $\Mkp$
with the adjacency relation defined in the same manner as for $\Gzp$ (hence $\mathcal{G}_0(p)$ coincides with $\Gzp \sqcup \{(0,0,0)\}$, where $(0,0,0)$ is an isolated vertex).
The graph structure of $\Gkp$ depends sensitively on $\kappa$ and $p$ in general.
For example, in the case of $\kappa=4$ (corresponding to a degenerate surface called the {\it Cayley cubic}, cf. \cite{CL2020}), the graph $\mathcal{G}_4(p)$ possesses many small components.
While $\Gkp$ is no longer connected in general,
it is still expected that for $\kappa\neq 4$,
there exists a unique {\it giant component} $\mathcal{C}_\kappa(p)$ in
$\Gkp$ such that $\#(\Gkp \setminus \mathcal{C}_\kappa(p))=o_\kappa(p^2)$. In a recent preprint~\cite{Mart2025}, Martin announced that this in fact holds for a majority of primes.
Moreover, the main result of \cite{Mart2025} would imply that $\#(\Gkp \setminus \mathcal{C}_\kappa(p))=O(1)$, representing a significant advance in the understanding of the global structure of $\Gkp$.
However, the precise structure of $\Gkp$ (and $\mathcal{C}_\kappa(p)$) is still largely unexplored.

In this paper, we investigate the topological properties of the generalized Markoff mod $p$ graph $\Gkp$ by combining perspectives of graph theory, computer algebra, and number theory.
In particular, by leveraging computational algebraic methods including resultants and Gr\"obner bases, in conjunction with the Chebotarev density theorem, we derive the following results based on graph minor theory.
Throughout this paper, the term {\it $K_{3,3}$-subdivision} refers to a subdivision of the complete bipartite graph $K_{3,3}$.

In this paper, we investigate the topological properties of the generalized Markoff mod $p$ graph $\Gkp$ by combining perspectives of graph theory, computer algebra, and number theory.
In particular, by leveraging computational algebraic methods including resultants and Gr\"obner bases, in conjunction with the Chebotarev density theorem, we derive the following results based on graph minor theory.
Throughout this paper, the term {\it $K_{3,3}$-subdivision} refers to a subdivision of the complete bipartite graph $K_{3,3}$.

Theorem~\ref{thm:main} yields several topological properties of $\Gkp$.
In particular, the following corollary is an immediate consequence of Theorem~\ref{thm:main}.

\begin{theorem}[Theorem~\ref{thm-disjoint-k}]
\label{thm:disjoint-intro}
For each $\kappa \in \mathbb{Z} \setminus \{ 4\}$, $\Gkp$ contains at least four mutually vertex-disjoint $K_{3,3}$-subdivisions for infinitely many primes $p$ (depending on $\kappa$) of natural density at least $1/4$ if $\kappa=2$ and $1/2$ otherwise.
Moreover, for almost all $\kappa$, the natural density of primes $p$ for which this holds is at least $5/8$.
\end{theorem}

\begin{theorem}
\label{thm:n=2 non-planarity}
Let $\kappa\in\mathbb{Z}\setminus\{4\}$.
Then, there exists an explicit $K_{3,3}$-subdivision in $\Gkp$ if $\xi_{2,\kappa}\eta_{2,\kappa}\ne 0$, $5(\kappa^2-73\kappa+61)\ne 0$,
and either of the following conditions holds.
\begin{itemize}
\item[$\mathrm{(A)}$]
$\left(\dfrac{\eta_{2,\kappa}}{p}\right)=1$ and
$\left(\dfrac{5}{p}\right)
=\left(\dfrac{8\kappa-17+2\sqrt{\eta_{2,\kappa}}}{p}\right)
=\left(\dfrac{8\kappa-17-2\sqrt{\eta_{2,\kappa}}}{p}\right)=1$, or,
\\[0pt]
\item[$\mathrm{(B)}$]
$\left(\dfrac{\eta_{2,\kappa}}{p}\right)=-1$ and $\left(\dfrac{5}{p}\right)=1$.
\end{itemize}
\end{theorem}

\subsection{Natural densities}
\label{subsec:density}
Let $\kappa\in\mathbb{Z}\setminus\{4\}$.
From Theorem~\ref{thm-n=1}, Theorem~\ref{thm:n=2 non-planarity} (B),
or Theorem~\ref{thm-n=(p-1)/2},
it follows from Dirichlet's theorem on arithmetic progressions that there are infinitely many primes $p$ for which
$\Gkp$ contains a $K_{3,3}$-subdivision, which yields topological properties of $\Gkp$ such as the non-planarity and existence of short cycles by Corollaries~\ref{cor:sufficient condition non-planarity} and \ref{cor:cycles}, respectively.
This motivates us to consider the natural density of such primes $p$: \[
\delta_{\kappa}
\coloneq\lim_{x\to\infty}
\frac{\#\{p\le x\,|\,\text{$\Gkp$ contains a $K_{3,3}$-subdivision}\}}{\#\{p\le x\}}.
\]
It holds that $\delta_0=1$
since $\mathcal{G}(p)$ is non-planar for any $p\neq 7$
by \cite[p.115]{C2024}.
For any $\kappa \neq 4$, Theorem~\ref{thm-n=1} and the Chebotarev density theorem (see, e.g.~\cite{N1999})
imply that $ \delta_{\kappa}\geq 1/2$.
Moreover, combining the three theorems above,
we arrive at the following theorem.

\begin{theorem}
\label{thm:np density}
\begin{itemize}
\item[$(1)$]
For each $\kappa\in\mathbb{Z}\setminus\{4\}$ such that none of
$\eta_{1,\kappa}$,
$\eta_{2,\kappa}$,
$5$, and $\kappa-4$
is a square in $\mathbb{Z}$,
and that the ratio of any two of them is not a square in $\mathbb{Q}$,
we have
\begin{equation}
\label{for:lower bound of Dnp}
\delta_{\kappa}\ge \frac{13}{16}.
\end{equation}
That is, $\Gkp$ contains a $K_{3,3}$-subdivision
for at least $81.25\%$
of primes $p$ asymptotically.
\item[$(2)$]
The natural density
of $\kappa\in\mathbb{Z}\setminus\{4\}$ for which $(1)$ holds
is equal to $1$.
\end{itemize}
\end{theorem}

\begin{theorem}
\label{thm-disjoint-k}
\begin{enumerate}
\item[$(1)$]
For each $\kappa\in\mathbb{Z}\setminus\{4\}$,
$\Gkp$ contains at least four mutually vertex-disjoint
$K_{3,3}$-subdivisions for infinitely many primes $p$ of natural density at least $1/4$ if $\kappa=2$ and $1/2$ otherwise.
Furthermore, if $\kappa$ is an integer such that
none of $\eta_{1, \kappa}$, $\eta_{2,\kappa}$ and $5$ is a square in $\mathbb{Z}$ and that the ratio of any two of them is also not a square in $\mathbb{Q}$,
then the natural density of such primes is at least $5/8$.
\item[$(2)$]
The natural density of $\kappa\in\mathbb{Z}\setminus\{4\}$
for which the latter assertion of $(1)$ holds is equal to $1$.
\end{enumerate}
\end{theorem}

\begin{remark}
The four mutually vertex-disjoint $K_{3,3}$-subdivisions above are not necessarily contained in the giant component $\mathcal{C}_{\kappa}(p)$ in general.
Indeed, for $\kappa=2$ and $3$, then
the $K_{3,3}$-subdivision $K\in \mathcal{K}^{\mathrm{prop}}_{1,\kappa}(p)$
obtained from Theorem~\ref{thm-n=1} (along with its three images under double sign changes) is contained in $\mathcal{S}_{2}(p)$ and $\mathcal{S}_{3}(p)$, respectively, where $\mathcal{S}_{2}(p)$ is a unique $16$-vertex component in Remark~\ref{rem-k=2} and $\mathcal{S}_{3}(p)$ denotes a unique $72$-vertex component in $\mathcal{G}_3(p)$ corresponds to the orbit of $((1+\sqrt{5})/2, 1, 1) \in \mathcal{M}_{3}(p)$
under the group generated by Vieta involutions,
provided that $(\frac{5}{p})=1$
(see Figure~\ref{fig:k3smallcomponent}).
On the other hand, according to a recent preprint \cite{Mart2025}, for $\kappa\in \mathbb{Z} \setminus \{2, 3, 4\}$,
all the constructed $K_{3,3}$-subdivisions are expected to be contained within $\mathcal{C}_\kappa(p)$ of $\Gkp$.
Furthermore, even in the cases $\kappa=2$ and $3$,
the $K_{3,3}$-subdivision with 24 vertices constructed in Theorem~\ref{thm:n=2 non-planarity} $(n=2)$ must be contained in $\mathcal{C}_\kappa(p)$.
This follows from the proof of Theorem~\ref{thm-disjoint-k}, together with a simple observation that for $\kappa=2$ and $3$,
any component other than $\mathcal{C}_\kappa(p)$ has at most $72$ vertices
and hence lacks sufficient space to accommodate all four mutually vertex-disjoint copies of the subdivision.
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.9\linewidth, height=0.2
\textheight, keepaspectratio]{Small_component_k=3_colored.pdf}
\caption{The $72$-vertex component $\mathcal{S}_3(p)$ of $\mathcal{G}_3(p)$ containing four mutually vertex-disjoint $K_{3,3}$-subdivisions as established in Theorem~\ref{thm-disjoint-k}}
\label{fig:k3smallcomponent}
\end{center}
\end{figure}
\end{remark}

Theorem~\ref{thm:main} yields several topological properties of $\Gkp$.
In particular, the following corollary is an immediate consequence of Theorem~\ref{thm:main}.

\begin{cor}
\label{cor:direct}
Under the same assumptions on $\kappa$ and $p$ of Theorem~\ref{thm:main} $(1)$,
the followings hold:
\begin{enumerate}
\item[$(1)$]
$(\mathrm{Corollary}~\ref{cor:sufficient condition non-planarity})$
$\Gkp$ is non-planar.
\item[$(2)$]
$(\mathrm{Corollary}~\ref{cor:cycles})$
$\Gkp$ has cycles (i.e. closed paths with no repeated vertices or edges) of lengths $6,9,10,15$, and $18$,
all of which can be constructed explicitly.
\end{enumerate}
\end{cor}
Note that it would be interesting to obtain results analogous to Theorem~\ref{thm:main} and Corollary~\ref{cor:direct} for other generalizations of the Markoff mod $p$ graph $\Gzp$ (e.g. \cite{CLM2025}); these will be discussed in our subsequent works.

Recall that $\Gkp$ admits the following non-trivial graph automorphisms, known as {\it double sign changes}:
\[
(x,y,z) \mapsto (x,-y,-z), \quad
(x,y,z) \mapsto (-x,y,-z), \quad \text{and} \quad
(x,y,z) \mapsto (-x,-y,z).
\]
These automorphisms, together with the identity, form a group isomorphic to $(\mathbb{Z}/2\mathbb{Z})^2$.
Combining this symmetry of $\Gkp$
with Theorem~\ref{thm:main},
we obtain an explicit construction of mutually vertex-disjoint $K_{3,3}$-subdivisions.

\begin{theorem}[Theorem~\ref{thm-disjoint-k}]
\label{thm:disjoint-intro}
For each $\kappa \in \mathbb{Z} \setminus \{ 4\}$, $\Gkp$ contains at least four mutually vertex-disjoint $K_{3,3}$-subdivisions for infinitely many primes $p$ (depending on $\kappa$) of natural density at least $1/4$ if $\kappa=2$ and $1/2$ otherwise.
Moreover, for almost all $\kappa$, the natural density of primes $p$ for which this holds is at least $5/8$.
\end{theorem}

Theorem~\ref{thm:disjoint-intro} strengthens Corollary~\ref{cor:direct} (1). Consider an embedding of $\Gkp$ into a closed surface (i.e. a compact topological surface without boundary); see e.g.~\cite{GT1987, MT2001} for the details on surface embeddings of graphs.
Recall that the Euler characteristic $\chi$ of $\Gkp$ is defined by $V-E+F$, where $V, E$, and $F$
denote the numbers of vertices, edges and faces, respectively.
Combining \cite[Theorem~1.1]{Mart2025} with Theorem~\ref{thm:disjoint-intro},
it follows that $\chi\leq -6$
for $\kappa$ and $p$ as in Theorem~\ref{thm:disjoint-intro},
whereas Euler's formula requires  $\chi=2$ if $\Gkp$ is planar.
Moreover, the existence of mutually vertex-disjoint $K_{3,3}$-subdivisions highlights the robustness of the non-planarity of $\Gkp$ in the sense that, by Theorem~\ref{thm:disjoint-intro}, one must remove at least four vertices to make the graph planar.
It is worth noting that, in light of \cite[Theorem~1.1]{Mart2025}, our construction of $K_{3,3}$-subdivisions implies that the Euler characteristic of the giant component $\mathcal{C}_\kappa(p)$ is at most $-6$ for any $\kappa\neq 4$
and infinitely many primes $p$ (see Section~\ref{sect:disjoint} for details).
From these structural perspectives, our results serve as heuristic evidence for the expander property of the giant component $\mathcal{C}_\kappa(p)$.
At least, $\mathcal{C}_\kappa(p)$ in fact possesses certain complex structure similar to that of $\Gzp$.

It is also worth mentioning that in cryptography and number theory, studying the structure of $\Gkp$ is crucial for understanding the security of the associated cryptographic hash functions proposed in \cite{FLLT2021, Sil2025}, which are expected to be expander hash functions~\cite{CLG2009}.
In particular, it is a challenging open problem to investigate the existence and distribution of short cycles in $\Gkp$, as these properties directly affect the collision resistance of the hash functions (\cite{D2023, FLLT2021}).
Corollary~\ref{cor:direct} (2) provides a modest step toward addressing this problem.

In addition, for $\kappa=0$,
we give a complete characterization for primes $p$ for which the graph $\Gzp$ admits an embedding into closed surfaces of minimal topological complexity beyond the plane.
\begin{theorem}[Theorem~\ref{thm-proj}]
\label{thm:embedded-intro}
Let $p>3$ be a prime. Then the graph $\Gzp$ admits an embedding into the torus
(i.e. the orientable closed surface of Euler characteristic zero)
or the (real) projective plane
(i.e. the non-orientable closed surface of Euler characteristic one)
if and only if $p=7$.
\end{theorem}

This result is achieved through Theorem~\ref{thm:main} by finding suitable forbidden minors for these embeddings, as well as the numerical verification of the connectivity of $\Gzp$ for $p<10^6$ (\cite{B2025}).
Note that the graph $\mathcal{G}(7)$ is planar, and hence one can draw it on any closed surface without edge crossings.
Consequently, Theorem~\ref{thm:embedded-intro} immediately implies the non-planarity of $\Gzp$ for any $p\neq 7$, and hence strengthens the main result of \cite{C2024}.

The rest of this paper is organized as follows.
Section~\ref{sect:chebyshev} introduces the basic and relevant properties of  Chebyshev polynomials, which play a key role in constructing $K_{3, 3}$-subdivisions in $\Gkp$.
In Section~\ref{sec:Six triples}, we present a systematic construction of $K_{3,3}$-subdivisions based on careful computations of Chebyshev polynomials and their associated resultants.
Briefly, we first reduce the construction to the problem of solving systems of algebraic equations via Gr\"obner bases to obtain candidates for the sextuples of triples that form the desired subdivisions.
Then we establish sufficient conditions for the ``properness'', which guarantees the claimed topological properties, of the subdivisions corresponding to the obtained sextuples.
Based on these arguments, Section~\ref{sect:non-planarity} provides explicit $K_{3,3}$-subdivisions.
Finally, Sections~\ref{sect:disjoint} and \ref{sect:embedded} are devoted to the proofs of Theorems~\ref{thm:disjoint-intro} and \ref{thm:embedded-intro}, respectively.

The introduction outlines the overall strategy leading to Theorem~\ref{thm:main} as follows. Chebyshev polynomials are introduced as a key tool: they “play a key role in constructing $K_{3, 3}$-subdivisions in $\Gkp$.” The construction is then made systematic via “careful computations of Chebyshev polynomials and their associated resultants.” More specifically, “we first reduce the construction to the problem of solving systems of algebraic equations via Gr\"obner bases to obtain candidates for the sextuples of triples that form the desired subdivisions.” After producing these candidates, the next step is to “establish sufficient conditions for the \`\`properness\'\', which guarantees the claimed topological properties, of the subdivisions corresponding to the obtained sextuples.” Using these arguments, the paper then “provides explicit $K_{3,3}$-subdivisions,” yielding Theorem~\ref{thm:main}.

The introduction outlines the overall strategy leading to the main theorem as follows. Chebyshev polynomials are introduced as a key tool: they “play a key role in constructing $K_{3, 3}$-subdivisions in $\Gkp$.” The construction is then made systematic via “careful computations of Chebyshev polynomials and their associated resultants.” More specifically, “we first reduce the construction to the problem of solving systems of algebraic equations via Gr\"obner bases to obtain candidates for the sextuples of triples that form the desired subdivisions.” After producing these candidates, the next step is to “establish sufficient conditions for the \`\`properness\'\'”, which guarantees the claimed topological properties, of the subdivisions corresponding to the obtained sextuples.” Using these arguments, the paper then “provides explicit $K_{3,3}$-subdivisions,” thereby establishing the main theorem.

\label{thm:main}
Let $\kappa \in\mathbb{Z}\setminus \{4\}$.
Then the followings hold:
\begin{enumerate}
\item[$(1)$]
The following three theorems provide sufficient conditions for the existence of an explicit $K_{3,3}$-subdivision in $\Gkp$:
\begin{theorem}
\label{thm-n=1}
Let $\kappa\in\mathbb{Z}\setminus\{4\}$.
Then, there exists an explicit $K_{3,3}$-subdivision in $\Gkp$
if $\xi_{1,\kappa}\eta_{1,\kappa}\ne 0$ and $\left(\frac{\eta_{1,\kappa}}{p}\right)=1$.
\end{theorem}
\begin{theorem}
\label{thm:n=2 non-planarity}
Let $\kappa\in\mathbb{Z}\setminus\{4\}$.
Then, there exists an explicit $K_{3,3}$-subdivision in $\Gkp$ if $\xi_{2,\kappa}\eta_{2,\kappa}\ne 0$, $5(\kappa^2-73\kappa+61)\ne 0$,
and either of the following conditions holds.
\begin{itemize}
\item[$\mathrm{(A)}$]
$\left(\dfrac{\eta_{2,\kappa}}{p}\right)=1$ and
$\left(\dfrac{5}{p}\right)
=\left(\dfrac{8\kappa-17+2\sqrt{\eta_{2,\kappa}}}{p}\right)
=\left(\dfrac{8\kappa-17-2\sqrt{\eta_{2,\kappa}}}{p}\right)=1$, or,
\\[0pt]
\item[$\mathrm{(B)}$]
$\left(\dfrac{\eta_{2,\kappa}}{p}\right)=-1$ and $\left(\dfrac{5}{p}\right)=1$.
\end{itemize}
\end{theorem}
\begin{theorem}
\label{thm-n=(p-1)/2}
Let $\kappa\in\mathbb{Z}\setminus\{4\}$.
Then, there exists an explicit $K_{3,3}$-subdivision in $\Gkp$ if $\left(\frac{\kappa-4}{p}\right)=1$.
\end{theorem}
For each such $\kappa$ and for infinitely many primes $p$ (depending on $\kappa$) of natural density at least $1/2$,
there exists an explicit $K_{3,3}$-subdivision in $\Gkp$.

\item[$(2)$]
Moreover, the following theorem implies that for almost all $\kappa$,
the natural density of primes $p$ for which $(1)$ holds is at least $13/16$.
\begin{theorem}
\label{thm:np density}
\begin{itemize}
\item[$(1)$]
For each $\kappa\in\mathbb{Z}\setminus\{4\}$ such that none of
$\eta_{1,\kappa}$,
$\eta_{2,\kappa}$,
$5$, and $\kappa-4$
is a square in $\mathbb{Z}$,
and that the ratio of any two of them is not a square in $\mathbb{Q}$,
we have
\begin{equation}
\label{for:lower bound of Dnp}
\delta_{\kappa}\ge \frac{13}{16}.
\end{equation}
That is, $\Gkp$ contains a $K_{3,3}$-subdivision
for at least $81.25\%$
of primes $p$ asymptotically.
\item[$(2)$]
The natural density
of $\kappa\in\mathbb{Z}\setminus\{4\}$ for which $(1)$ holds
is equal to $1$.
\end{itemize}
\end{theorem}

\end{enumerate},

Question: Let \(\kappa\in\mathbb Z\setminus\{4\}\). For a prime \(p>3\), let \(\mathcal M_\kappa(p)\subset \mathbb F_p^3\) be the set of solutions to \(x^2+y^2+z^2=xyz+\kappa\), and let \(\mathcal G_\kappa(p)\) be the graph with vertex set \(\mathcal M_\kappa(p)\), where two vertices are adjacent exactly when one is obtained from the other by a Vieta involution \(R_1(x,y,z)=(yz-x,y,z)\), \(R_2(x,y,z)=(x,zx-y,z)\), or \(R_3(x,y,z)=(x,y,xy-z)\). Write \(\left(\frac{\cdot}{p}\right)\) for the Legendre symbol, and let \(\xi_{1,\kappa},\eta_{1,\kappa},\xi_{2,\kappa},\eta_{2,\kappa}\) denote the auxiliary integers attached to \(\kappa\). Which statement about subdivisions of \(K_{3,3}\) in \(\mathcal G_\kappa(p)\) holds?

An explicit \(K_{3,3}\)-subdivision exists in \(\mathcal G_\kappa(p)\) whenever at least one of the following holds: (i) \(\xi_{1,\kappa}\eta_{1,\kappa}\neq 0\) and \(\left(\frac{\eta_{1,\kappa}}{p}\right)=1\); (ii) \(\xi_{2,\kappa}\eta_{2,\kappa}\neq 0\), \(5(\kappa^2-73\kappa+61)\neq 0\), and either (A) \(\left(\frac{\eta_{2,\kappa}}{p}\right)=1\) and \(\left(\frac{5}{p}\right)=\left(\frac{8\kappa-17+2\sqrt{\eta_{2,\kappa}}}{p}\right)=\left(\frac{8\kappa-17-2\sqrt{\eta_{2,\kappa}}}{p}\right)=1\), or (B) \(\left(\frac{\eta_{2,\kappa}}{p}\right)=-1\) and \(\left(\frac{5}{p}\right)=1\); (iii) \(\left(\frac{\kappa-4}{p}\right)=1\). Consequently, for every \(\kappa\in\mathbb Z\setminus\{4\}\), the set of primes \(p\) for which \(\mathcal G_\kappa(p)\) contains an explicit \(K_{3,3}\)-subdivision is infinite and has natural density at least \(1/2\). Moreover, if none of \(\eta_{1,\kappa},\eta_{2,\kappa},5,\kappa-4\) is a square in \(\mathbb Z\) and the ratio of any two of them is not a square in \(\mathbb Q\), then the set of primes \(p\) for which \(\mathcal G_\kappa(p)\) contains a \(K_{3,3}\)-subdivision has natural density at least \(13/16\) (equivalently, at least \(81.25\%\) of primes asymptotically), and the set of integers \(\kappa\in\mathbb Z\setminus\{4\}\) satisfying this arithmetic hypothesis has natural density \(1\).

Choices:
B. An explicit \(K_{3,3}\)-subdivision exists in \(\mathcal G_\kappa(p)\) whenever at least one of the following holds: (i) \(\xi_{1,\kappa}\eta_{1,\kappa}\neq 0\) and \(\left(\frac{\eta_{1,\kappa}}{p}\right)=1\); (ii) \(\xi_{2,\kappa}\eta_{2,\kappa}\neq 0\), \(5(\kappa^2-73\kappa+61)\neq 0\), and either (A) \(\left(\frac{\eta_{2,\kappa}}{p}\right)=1\) and \(\left(\frac{5}{p}\right)=\left(\frac{8\kappa-17+2\sqrt{\eta_{2,\kappa}}}{p}\right)=\left(\frac{8\kappa-17-2\sqrt{\eta_{2,\kappa}}}{p}\right)=1\), or (B) \(\left(\frac{\eta_{2,\kappa}}{p}\right)=-1\) and \(\left(\frac{5}{p}\right)=-1\); (iii) \(\left(\frac{\kappa-4}{p}\right)=1\). Consequently, for every \(\kappa\in\mathbb Z\setminus\{4\}\), the set of primes \(p\) for which \(\mathcal G_\kappa(p)\) contains an explicit \(K_{3,3}\)-subdivision is infinite and has natural density at least \(1/2\). Moreover, if none of \(\eta_{1,\kappa},\eta_{2,\kappa},5,\kappa-4\) is a square in \(\mathbb Z\) and the ratio of any two of them is not a square in \(\mathbb Q\), then the set of primes \(p\) for which \(\mathcal G_\kappa(p)\) contains a \(K_{3,3}\)-subdivision has natural density at least \(13/16\), and the set of integers \(\kappa\in\mathbb Z\setminus\{4\}\) satisfying this arithmetic hypothesis has natural density \(1\).
C. If at least one of the following holds for a prime \(p>3\): (i) \(\xi_{1,\kappa}\eta_{1,\kappa}\neq 0\) and \(\left(\frac{\eta_{1,\kappa}}{p}\right)=1\); (ii) \(\xi_{2,\kappa}\eta_{2,\kappa}\neq 0\), \(5(\kappa^2-73\kappa+61)\neq 0\), and either (A) \(\left(\frac{\eta_{2,\kappa}}{p}\right)=1\) together with \(\left(\frac{5}{p}\right)=\left(\frac{8\kappa-17+2\sqrt{\eta_{2,\kappa}}}{p}\right)=\left(\frac{8\kappa-17-2\sqrt{\eta_{2,\kappa}}}{p}\right)=1\), or (B) \(\left(\frac{\eta_{2,\kappa}}{p}\right)=-1\) and \(\left(\frac{5}{p}\right)=1\); or (iii) \(\left(\frac{\kappa-4}{p}\right)=1\), then \(\mathcal G_\kappa(p)\) contains an explicit \(K_{3,3}\)-subdivision. Consequently, for every \(\kappa\in\mathbb Z\setminus\{4\}\), there are infinitely many primes \(p\) for which \(\mathcal G_\kappa(p)\) contains an explicit \(K_{3,3}\)-subdivision.
D. An explicit \(K_{3,3}\)-subdivision exists in \(\mathcal G_\kappa(p)\) whenever at least one of the following holds: (i) \(\xi_{1,\kappa}\eta_{1,\kappa}\neq 0\) and \(\left(\frac{\eta_{1,\kappa}}{p}\right)=1\); (ii) \(\xi_{2,\kappa}\eta_{2,\kappa}\neq 0\), \(5(\kappa^2-73\kappa+61)\neq 0\), and \(\left(\frac{\eta_{2,\kappa}}{p}\right)=1\) with \(\left(\frac{5}{p}\right)=\left(\frac{8\kappa-17+2\sqrt{\eta_{2,\kappa}}}{p}\right)=\left(\frac{8\kappa-17-2\sqrt{\eta_{2,\kappa}}}{p}\right)=1\); (iii) \(\left(\frac{\kappa-4}{p}\right)=1\). Consequently, for every \(\kappa\in\mathbb Z\setminus\{4\}\), the set of primes \(p\) for which \(\mathcal G_\kappa(p)\) contains an explicit \(K_{3,3}\)-subdivision is infinite and has natural density at least \(1/2\). Moreover, if none of \(\eta_{1,\kappa},\eta_{2,\kappa},5,\kappa-4\) is a square in \(\mathbb Z\) and the ratio of any two of them is not a square in \(\mathbb Q\), then the set of primes \(p\) for which \(\mathcal G_\kappa(p)\) contains a \(K_{3,3}\)-subdivision has natural density at least \(13/16\), and the set of integers \(\kappa\in\mathbb Z\setminus\{4\}\) satisfying this arithmetic hypothesis has natural density \(1\).
E. An explicit \(K_{3,3}\)-subdivision exists in \(\mathcal G_\kappa(p)\) whenever at least one of the following holds: (i) \(\xi_{1,\kappa}\eta_{1,\kappa}\neq 0\) and \(\left(\frac{\eta_{1,\kappa}}{p}\right)=1\); (ii) \(\xi_{2,\kappa}\eta_{2,\kappa}\neq 0\), \(5(\kappa^2-73\kappa+61)\neq 0\), and either (A) \(\left(\frac{\eta_{2,\kappa}}{p}\right)=1\) and \(\left(\frac{5}{p}\right)=\left(\frac{8\kappa-17+2\sqrt{\eta_{2,\kappa}}}{p}\right)=\left(\frac{8\kappa-17-2\sqrt{\eta_{2,\kappa}}}{p}\right)=1\), or (B) \(\left(\frac{\eta_{2,\kappa}}{p}\right)=-1\) and \(\left(\frac{5}{p}\right)=1\); (iii) \(\left(\frac{\kappa-4}{p}\right)=1\). Consequently, for every \(\kappa\in\mathbb Z\setminus\{4\}\), the set of primes \(p\) for which \(\mathcal G_\kappa(p)\) contains an explicit \(K_{3,3}\)-subdivision is infinite and has natural density at least \(1/2\). Moreover, if none of \(\eta_{1,\kappa},\eta_{2,\kappa},5,\kappa-4\) is a square in \(\mathbb Z\) and the ratio of any two of them is not a square in \(\mathbb Q\), then the set of primes \(p\) for which \(\mathcal G_\kappa(p)\) contains a \(K_{3,3}\)-subdivision has natural density at least \(3/4\), and the set of integers \(\kappa\in\mathbb Z\setminus\{4\}\) satisfying this arithmetic hypothesis has natural density \(1\).

lower-density bound 13/16 replaced by 3/4