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arXiv · 2205.08564

The overfull conjecture on graphs of odd order and large minimum degree

Songling Shan
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Abstract

Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor \frac{1}{2}|V(H)| \rfloor$. Chetwynd and Hilton in 1986 conjectured that a graph $G$ with $\Delta(G)>\frac{1}{3}|V(G)|$ has chromatic index $\Delta(G)$ if and only if $G$ contains no overfull subgraph. Let $0<\varepsilon <1$ and $G$ be a large graph on $n$ vertices with minimum degree at least $\frac{1}{2}(1+\varepsilon)n$. It was shown that the conjecture holds for $G$ if $n$ is even. In this paper, the same result is proved if $n$ is odd. As far as we know, this is the first result on the conjecture for graphs of odd order and with a minimum degree constraint.

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Songling Shan. 2022-05-17. The overfull conjecture on graphs of odd order and large minimum degree. https://arxiv.org/abs/2205.08564

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