arXiv subjects

Leyou Xu

Publications and source records attributed to Leyou Xu.

23 records · Page 2Linked to original sources

arXiv2023

Toughness and existence of $2$-factors

A graph is $t$-tough if the deletion of any set of, say, $m$ vertices from the graph leaves a graph with at most $\frac{m}{t}$ components. In 1973, Chvátal suggested the problem of relating toughness to factors in graphs. In 1985, Enomoto et al. showed that each $2$-tough graph with at least three vertices has a $2$-factor, but for any $ε>0$, there exists a $(2-ε)$-tough graph on at least $3$ vertices having no $2$-factor. In recent years, the study of sufficient conditions for graphs with toughness less than $2$ having a $2$-factor has received a paramount interest. In this paper, we give new tight sufficient conditions for a $t$-tough graph having a $2$-factor when $1\le t<2$ by involving independence number, minimum degree, connectivity and forbidden forests.

math.CO↗

arXiv2023

Laplacain eigenvalue distribution and diameter of graphs

Let $G$ be a connected graph on $n$ vertices with diameter $d$. It is known that if $2\le d\le n-2$, there are at most $n-d$ Laplacian eigenvalues in the interval $[n-d+2, n]$. In this paper, we show that if $1\le d\le n-3$, there are at most $n-d+1$ Laplacian eigenvalues in the interval $[n-d+1, n]$. Moreover, we try to identify the connected graphs on $n$ vertices with diameter $d$, where $2\le d\le n-3$, such that there are at most $n-d$ Laplacian eigenvalues in the interval $[n-d+1, n]$.

math.CO↗

arXiv2023

Proof of a conjecture on distribution of Laplacian eigenvalues and diameter, and beyond

Ahanjideh, Akbari, Fakharan and Trevisan proposed a conjecture in [Linear Algebra Appl. 632 (2022) 1--14] on the distribution of the Laplacian eigenvalues of graphs: for any connected graph of order $n$ with diameter $d\ge 2$ that is not a path, the number of Laplacian eigenvalues in the interval $[n-d+2,n]$ is at most $n-d$. We show that the conjecture is true, and give a complete characterization of graphs for which the conjectured bound is attained. This establishes an interesting relation between the spectral and classical parameters.

math.CO↗

arXiv2023

The generalized $4$-connectivity of bubble-sort graphs

For $S\subseteq V(G)$ with $|S|\ge 2$, let $κ_G (S)$ denote the maximum number of internally disjoint trees connecting $S$ in $G$. For $2\le k\le n$, the generalized $k$-connectivity $κ_k(G)$ of an $n$-vertex connected graph $G$ is defined to be $κ_k(G)=\min \{κ_G(S): S\in V(G) \mbox{ and } |S|=k\}$. The generalized $k$-connectivity can serve for measuring the fault tolerance of an interconnection network. The bubble-sort graph $B_n$ for $n\ge 2$ is a Cayley graph over the symmetric group of permutations on $[n]$ generated by transpositions from the set $\{[1,2],[2,3],\dots, [n-1,n]\}$. In this paper, we show that for the bubble-sort graphs $B_n$ with $n\ge 3$, $κ_4(B_n)=n-2$.

math.CO↗

arXiv2023

Hamiltonicity of $1$-tough $(P_2\cup kP_1)$-free graphs

Given a graph $H$, a graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. For a positive real number $t$, a non-complete graph $G$ is said to be $t$-tough if for every vertex cut $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. A complete graph is said to be $t$-tough for any $t>0$. Chvátal's toughness conjecture, stating that there exists a constant $t_0$ such that every $t_0$-tough graph with at least three vertices is Hamiltonian, is still open in general. Chvátal and Erdös \cite{CE} proved that, for any integer $k\ge 1$, every $\max\{2,k\}$-connected $(k+1)P_1$-free graph on at least three vertices is Hamiltonian. Along the Chvátal-Erdös theorem, Shi and Shan \cite{SS} proved that, for any integer $k\ge 4$, every $4$-tough $2k$-connected $(P_2\cup kP_1)$-free graph with at least three vertices is Hamiltonian, and furthermore, they proposed a conjecture that for any integer $k\ge 1$, any $1$-tough $2k$-connected $(P_2\cup kP_1)$-free graph is Hamiltonian. In this paper, we confirm the conjecture, and furthermore, we show that if $k\ge 3$, then the condition `$2k$-connected' may be weakened to be `$2(k-1)$-connected'. As an immediate consequence, for any integer $k\ge 3$, every $(k-1)$-tough $(P_2\cup kP_1)$-free graph is Hamiltonian. This improves the result of Hatfield and Grimm \cite{HG}, stating that every $3$-tough $(P_2\cup 3P_1)$-free graph is Hamiltonian.

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