arXiv subjects

Zai Ping Lu

Publications and source records attributed to Zai Ping Lu.

9 recordsLinked to original sources

arXiv2026

Disjoint chorded cycles in a $2$-connected graph

A chorded cycle in a graph $G$ is a cycle containing an edge of $G$ that joins two nonconsecutive vertices of the cycle. In 2010, Gao and Qiao independently proved that a graph of order at least $4s$, in which the neighborhood union of any two nonadjacent vertices has at least $4s+1$ vertices, contains $s$ vertex-disjoint chorded cycles. In 2022, Gould raised a problem that asks whether increasing connectivity would improve the neighborhood union condition. In this paper, we solve the problem for $2$-connected graphs by proving that a $2$-connected graph of order at least $4s$, in which the neighborhood union of any two nonadjacent vertices has at least $4s$ vertices, contains $s$ vertex-disjoint chorded cycles. Moreover, the neighborhood-union bound $4s$ is sharp.

math.CO↗

arXiv2025

Enumerating Cayley digraphs on dihedral groups

This paper investigates the enumeration of Cayley digraphs, focusing on counting Cayley digraphs on dihedral groups up to CI-isomorphism. By leveraging the Cauchy-Frobenius Lemma and properties of automorphisms, we derive an explicit formula for the number of non-isomorphic Cayley digraphs on dihedral groups with DCI-property, particularly for the group $\mathrm{D}_{6p}$ with $p>3$ a prime. The enumeration involves detailed analysis of cycle numbers of automorphisms and their actions on the group elements, culminating in a precise count of non-isomorphic digraphs.

math.CO↗

arXiv2024

On Isomorphisms of Tetravalent Cayley Digraphs over Dihedral Groups

Let $m$ be a positive integer. A group $G$ is said to be an $m$-DCI-group or an $m$-CI-group if $G$ has the $k$-DCI property or $k$-CI property for all positive integers $k$ at most $m$, respectively. Let $G$ be a dihedral group of order $2n$ with $n\geq 3$. Qu and Yu proved that $G$ is an $m$-DCI-group or $m$-CI-group, for every $m\in \{1,2,3\}$, if and only if $n$ is odd. In this paper, it is shown that $G$ is a $4$-DCI-group if and only if $n$ is odd and not divisible by $9$, and $G$ is a $4$-CI-group if and only if $n$ is odd.

math.CO↗

arXiv2022

On 2-arc-transitive graphs of product action type

In this paper, we discuss the structural information about 2-arc-transitive (non-bipartite and bipartite) graphs of product action type. It is proved that a 2-arc-transitive graph of product action type requires certain restrictions on either the vertex-stabilizers or the valency. Based on the existence of some equidistant linear codes, a construction is given for 2-arc-transitive graphs of non-diagonal product action type, which produces several families of such graphs. Besides, a nontrivial construction is given for 2-arc-transitive bipartite graphs of diagonal product action type

math.CO↗

arXiv2022

On basic $2$-arc-transitive graphs

A connected graph $Γ=(V,E)$ of valency at least $3$ is called a basic $2$-arc-transitive graph if its full automorphism group has a subgroup $G$ with the following properties: (i) $G$ acts transitively on the set of $2$-arcs of $Γ$, and (ii) every minimal normal subgroup of $G$ has at most two orbits on $V$. In her papers [17,18], Praeger proved a connected $2$-arc-transitive graph of valency at least $3$ is a normal cover of some basic $2$-arc-transitive graph, and characterized the group-theoretic structures for basic $2$-arc-transitive graphs. Based on Praeger's theorems on $2$-arc-transitive graphs, this paper presents a further understanding on basic $2$-arc-transitive graphs.

math.CO↗

arXiv2021

Two-arc-transitive graphs of odd order -- II

It is shown that each subgroup of odd index in an alternating group of degree at least 10 has all insoluble composition factors to be alternating. A classification is then given of 2-arc-transitive graphs of odd order admitting an alternating group or a symmetric group. This is the second of a series of papers aiming towards a classification of 2-arc-transitive graphs of odd order.

math.CO↗

arXiv2021

Symmetric graphs of prime valency with a transitive simple group

A graph $\Ga=(V,E)$ is called a Cayley graph of some group $T$ if the automorphism group $\Aut(\Ga)$ contains a subgroup $T$ which acts on regularly on $V$. If the subgroup $T$ is normal in $\Aut(\Ga)$ then $\Ga$ is called a normal Cayley graph of $T$. Let $r$ be an odd prime. Fang et al. \cite{FMW} proved that, with a finite number of exceptions for finite simple group $T$, every connected symmetric Cayley graph of $T$ of valency $r$ is normal. In this paper, employing maximal factorizations of finite almost simple groups, we work out a possible list of those exceptions for $T$.

math.GR↗

arXiv2020

On edge-primitive graphs with soluble edge-stabilizers

A graph is edge-primitive if its automorphism group acts primitively on the edge set, and 2-arc-transitive if its automorphism group acts transitively on the set of 2-arcs. In this paper, we present a classification for those edge-primitive graphs which are 2-arc-transitive and have soluble edge-stabilizers.

math.CO↗

arXiv2019

On the automorphism groups of graphs with twice prime valency

A graph is edge-transitive if its automorphism group acts transitively on the edge set. In this paper, we investigate the automorphism groups of edge-transitive graphs of odd order and twice prime valency. Let $Γ$ be a connected graph of odd order and twice prime valency, and let $G$ be a subgroup of the automorphism group of $\Ga$. In the case where $G$ acts transitively on the edges and quasiprimitively on the vertices of $\Ga$, we prove that either $G$ is almost simple or $G$ is a primitive group of affine type. If further $G$ is an almost simple primitive group then, with two exceptions, the socle of $G$ acts transitively on the edges of $Γ$.

math.CO↗