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arXiv · 2105.05333

The overfullness of graphs with small minimum degree and large maximum degree

Yan Cao·Guantao Chen·Guangming Jing·Songling Shan
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Abstract

Given a simple graph $G$, denote by $Δ(G)$, $δ(G)$, and $χ'(G)$ the maximum degree, the minimum degree, and the chromatic index of $G$, respectively. We say $G$ is \emph{$Δ$-critical} if $χ'(G)=Δ(G)+1$ and $χ'(H)\le Δ(G)$ for every proper subgraph $H$ of $G$; and $G$ is \emph{overfull} if $|E(G)|>Δ\lfloor |V(G)|/2 \rfloor$. Since a maximum matching in $G$ can have size at most $\lfloor |V(G)|/2 \rfloor$, it follows that $χ'(G) = Δ(G) +1$ if $G$ is overfull. Conversely, let $G$ be a $Δ$-critical graph. The well known overfull conjecture of Chetwynd and Hilton asserts that $G$ is overfull provided $Δ(G) > |V(G)|/3$. In this paper, we show that any $Δ$-critical graph $G$ is overfull if $Δ(G) - 7δ(G)/4\ge(3|V(G)|-17)/4$.

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Yan Cao, Guantao Chen, Guangming Jing, Songling Shan. 2021-05-11. The overfullness of graphs with small minimum degree and large maximum degree. https://arxiv.org/abs/2105.05333

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