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arXiv 2008.08135math.CO
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具有小核心度的临界2类图的过满度

Overfullness of critical class 2 graphs with a small core degree

Yan Cao, Guantao Chen, Songling Shan

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摘要：

设$G$是一个简单图，$n$，$Δ(G)$和$χ' (G)$分别是$G$的阶，最大度和色指数.对于$G$的每个真子图$H$，如果$|E(G)|/\lfloor n/2\rfloor > Δ(G)$是$G$过满的，如果$χ'(H) < χ'(G)$是$G$临界的.显然，如果$G$过满，则$χ'(G) = Δ(G)+1$。$G$的核心，记为$G_Δ$，是由$G$的所有最大度顶点导出的子图。Hilton和Zhao证明了：对于任何具有$Δ(G) \ge 4$的临界2类图$G$，如果$G_Δ$的最大度至多为2，则$G$是过满的，从而给出了$Δ(G) > n/2 +1$ .证明了对任意临界2类图$G$，若$G_Δ$的最小度至多为2且$Δ(G) > n/2 +1$，则$G$是过满图.

英文摘要：

Let $G$ be a simple graph, and let $n$, $Δ(G)$ and $χ' (G)$ be the order, the maximum degree and the chromatic index of $G$, respectively. We call $G$ overfull if $|E(G)|/\lfloor n/2\rfloor > Δ(G)$, and critical if $χ'(H) < χ'(G)$ for every proper subgraph $H$ of $G$. Clearly, if $G$ is overfull then $χ'(G) = Δ(G)+1$. The core of $G$, denoted by $G_Δ$, is the subgraph of $G$ induced by all its maximum degree vertices. Hilton and Zhao conjectured that for any critical class 2 graph $G$ with $Δ(G) \ge 4$, if the maximum degree of $G_Δ$ is at most two, then $G$ is overfull, which in turn gives $Δ(G) > n/2 +1$. We show that for any critical class 2 graph $G$, if the minimum degree of $G_Δ$ is at most two and $Δ(G) > n/2 +1$, then $G$ is overfull.

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