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arXiv 1404.6299math.CO
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涉及大最大次数的Vizing二因子猜想

Vizing's 2-factor Conjecture Involving Large Maximum Degree

Guantao Chen, Songling Shan

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摘要：

设$G$是$n$阶连通简单图，$Δ(G)$和$χ'(G)$分别表示$G$的最大度和色指数. Vizing证明了$χ'(G)=Δ(G)$或$Δ(G)+1$。根据该结果，如果每个$e\in E(G)$均为$χ'(G)=Δ(G)+1$和$χ'(G-e)=Δ(G)$，则将$G$称为$Δ$ -关键。1968年，Vizing证明了如果$G$是$n$ -顶点$Δ$ -临界图，则独立数$α(G)\le n/2$ .此外，他强调，事实上，PDMATHAMM有一个2因子。Luo和Zhao证明了：如果$G$是一个$n$ -顶点$Δ$ -临界图，且具有$Δ(G)\ge n/2$，则$α(G)\le n/2$ .最近，他们证明了如果$G$是一个$n$ -顶点$Δ$ -临界图，且有$Δ(G)\ge 6n/7$，则$G$有一个哈密尔顿圈，因此$G$有一个2-因子。本文证明了：如果$G$是一个$n$ -顶点$Δ$ -临界图，且有$Δ(G)\ge n/2$，则$G$有2-因子.

英文摘要：

Let $G$ be a connected simple graph of order $n$ and let $Δ(G)$ and $χ'(G)$ denote the maximum degree and chromatic index of $G$, respectively. Vizing proved that $χ'(G)=Δ(G)$ or $Δ(G)+1$. Following this result, $G$ is called $Δ$-critical if $χ'(G)=Δ(G)+1$ and $χ'(G-e)=Δ(G)$ for every $e\in E(G)$. In 1968, Vizing conjectured that if $G$ is an $n$-vertex $Δ$-critical graph, then the independence number $α(G)\le n/2$. Furthermore, he conjectured that, in fact, $G$ has a 2-factor. Luo and Zhao showed that if $G$ is an $n$-vertex $Δ$-critical graph with $Δ(G)\ge n/2$, then $α(G)\le n/2$. More recently, they showed that if $G$ is an $n$-vertex $Δ$-critical graph with $Δ(G)\ge 6n/7$, then $G$ has a hamiltonian cycle, and so $G$ has a 2-factor. In this paper, we show that if $G$ is an $n$-vertex $Δ$-critical graph with $Δ(G)\ge n/2$, then $G$ has a 2-factor.

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