arXiv subjects

Huye Chen

Publications and source records attributed to Huye Chen.

8 recordsLinked to original sources

arXiv2026

Hamilton cycles of semisymmetric graphs of order $2p^3$

In light of Lov\'{a}sz's longstanding question on the existence of Hamilton paths in vertex-transitive graphs, Du and Yuan considered a natural variant: what if vertex-transitivity is relaxed, while a high degree of symmetry--specifically edge-transitivity--is retained? To investigate this, they studied semisymmetric graphs (i.e. regular, edge-transitive, but not vertex-transitive graphs) and showed that every connected semisymmetric graph of order $2pq$, where $p$ and $q$ are distinct primes, contains a Hamilton cycle. In this paper, it is shown that for any prime $p$, every connected semisymmetric graph of order $2p^3$ also contains a Hamilton cycle.

math.CO↗

arXiv2026

Isomorphism factorizations of the complete graph into Cayley graphs on CI-groups

Isomorphic factorizations of complete graphs originate from the seminal work of Frank Harary and collaborators, who initiated the systematic study of decompositions of complete graphs into pairwise isomorphic spanning subgraphs. In this paper, we investigate isomorphic factorizations of complete graphs into Cayley graphs on CI-groups. Let $\Gamma=Cay(G,S)$ denote the Cayley graph of finite group $G$. We obtain a necessary and sufficient condition on CI-group $G$ so that the complete graph on $|G|$ vertices can be edge-partitioned into $k$-copies of Cayley graph of the same CI-group $G$ each isomorphic to $Cay(G,S)$ for some inverse-closed subset $S\subset G\setminus\{1\}$. Further we give a construction of isomorphic factorizations of the complete graph into Cayley graphs on CI-group.

math.CO↗

arXiv2025

The Burness-Giudici Conjecture on Some Primitive Groups with Socle PSU(3,q)

Let $G$ be a transitive permutation group on $\Omega$ with two points $\alpha, \beta\in\Omega$ such that $G_{\alpha}\cap G_{\beta}=1$. The Saxl graph $\Sigma(G)$ of the pair $(G,\Omega)$ is the graph with vertex set $\Omega$, while two vertices $\alpha', \beta'$ are adjacent if and only if $G_{\alpha'}\cap G_{\beta'}=1$. It was conjectured by Burness and Giudici that the Saxl graph $\Sigma(G)$ of any primitive permutation group $G$ has the property that any two vertices have a common neighbor. We focused on proving the conjecture for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$, that is, those with $soc(G)\in \{PSL(2,q), PSU(3,q), Ree(q), Sz(q)\}$. The case of $soc(G)=PSL(2,q)$ has been published in two papers. This paper will address most cases where $soc(G)=PSU(3,q)$, with the exception of a particularly intricate configuration in which the point stabilizer contains $PSO(3,q)$. That specific configuration has been treated in a separate paper.

math.GR↗

arXiv2025

The Burness-Giudici Conjecture on Primitive Groups with Socle PSU(3,q)

Let $G$ be a transitive permutation group on a set $\Omega$, and suppose $G_{\alpha}\cap G_{\beta}=1$ for some distinct $\alpha, \beta\in\Omega$. The Saxl graph $\Sigma(G)$ of $(G, \Omega)$ is defined as the graph with vertex set $\Omega$, where two vertices $\alpha', \beta'$ are adjacent if and only if $G_{\alpha'}\cap G_{\beta'}=1$. Burness and Giudici conjectured that for every primitive permutation group $G$, its Saxl graph has the property that any two vertices share a common neighbor. We focus on proving the conjecture for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$; that is, $soc(G)\in \{PSL(2,q),PSU(3,q), Ree(q),Sz(q)\}$. The case $soc(G)=PSL(2,q)$ has been treated in two earlier papers. The purpose of the present paper is to settle the case $soc(G)=PSU(3,q)$. To finsh this work, we draw on methods from abstract- and permutation- group theory, finite unitary geometry, probabilistic approach, number theory (employing Weil's bound), and, most importantly, algebraic combinatorics, which provides us some key ideas.

math.GR↗

arXiv2025

The Burness-Giudici Conjecture on Primitive Groups with Socle $Ree(q)$ and $Sz(q)$

Let $G$ be a transitive permutation group on $\Omega$ containing two points $\alpha, \beta$ such that $G_{\alpha}\cap G_{\beta}=1$. The Saxl graph $\Sigma(G)$ of $(G, \Omega)$ is defined as the graph with vertex set $\Omega$, where two vertices $\alpha', \beta'$ are adjacent if and only if $G_{\alpha'}\cap G_{\beta'}=1$. Burness and Giudici conjectured that for any primitive permutation group $G$, its Saxl graph $\Sigma(G)$ satisfies the property that any two vertices share a common neighbor. We focused on proving this conjecture for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$; that is, groups with $soc(G)\in \{PSL(2,q), PSU(3,q), Ree(q), Sz(q)\}$. The case $soc(G)=PSL(2,q)$ has been published in two papers. In this paper, we treat the cases where $soc(G)\in\{Ree(q), Sz(q)\}$.

math.GR↗

arXiv2025

Hamilton Cycles In Vertex-Transitive Graphs of Order 10p

After long-term efforts, the Hamilton path (cycle) problem for connected vertex-transitive graphs of order $pq$ (where $p$ and $q$ are primes) was finally resolved in 2021, see [10]. Fifteen years ago, mathematicians began addressing this problem for graphs of order $2pq$. Among these studies, it was proved in 2012 (see [21]) that every connected vertex-transitive graph of order $10p$ (where $p \neq 7$ is a prime) contains a Hamilton path, with the exception of a family of graphs that was recently confirmed in [11]. In this paper, we achieve a further result: every connected vertex-transitive graph of order $10p$ (where $p$ is a prime) contains a Hamilton cycle, except for the truncation of the Petersen graph.

math.CO↗

arXiv2024

Second largest maximal cliques in small Paley graphs of square order

There is a conjecture that the second largest maximal cliques in Paley graphs of square order $P(q^2)$ have size $\frac{q+\epsilon}{2}$, where $q \equiv \epsilon \pmod 4$, and split into two orbits under the full group of automorphisms whenever $q \ge 25$ (a symmetric description for these two orbits is known). However, some extra second largest maximal cliques (of this size) exist in $P(q^2)$ whenever $q \in \{9,11,13,17,19,23\}$. In this paper we analyse the algebraic and geometric structure of the extra cliques.

math.CO↗

arXiv2023

On the Burness-Giudici Conjecture

Let $G$ be a permutation group on a set $Ω$. A subset of $Ω$ is a base for $G$ if its pointwise stabilizer in $G$ is trivial. By $b(G)$ we denote the size of the smallest base of $G$. Every permutation group with $b(G)=2$ contains some regular suborbits. It is conjectured by Burness-Giudici in [4] that every primitive permutation group $G$ with $b(G)=2$ has the property that if $α^g\not\in Γ$ then $Γ\cap Γ^g\neq \emptyset$, where $Γ$ is the union of all regular suborbits of $G$ relative to $α$. An affirmative answer of the conjecture has been shown for many sporadic simple groups and some alternative groups in [4], but it is still open for simple groups of Lie-type. The first candidate of infinite family of simple groups of Lie-type we should work on might be $PSL(2,q)$, where $q\geq 5$. In this manuscript, we show the correctness of the conjecture for all the primitive groups with socle $PSL(2,q)$, see Theorem $1.3$.

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