周长为 9 且没有较长奇数孔的图形是 3 可取的

Pub Date : 2024-04-11 DOI:10.1002/jgt.23101

Yan Wang, Rong Wu

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引用次数: 0

摘要

对于一个数，让表示有周长且没有长度大于的奇数洞的图族。吴、徐和徐猜想，在的每一个图都是 3 可容的。Chudnovsky 等人、Wu 等人和 Chen 分别证明了、和中的每个图都是 3 可容的。在本文中，我们证明了每个 in 的图都是 3 有利的。这证实了 Wu、Xu 和 Xu 的猜想。

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Graphs with girth 9 and without longer odd holes are 3-colourable

For a number $l \geq 2$ , let $G_{l}$ denote the family of graphs which have girth $2 l + 1$ and have no odd hole with length greater than $2 l + 1$ . Wu, Xu and Xu conjectured that every graph in $⋃_{l \geq 2} G_{l}$ is 3-colourable. Chudnovsky et al., Wu et al., and Chen showed that every graph in $G_{2}$ , $G_{3}$ and $⋃_{l \geq 5} G_{l}$ is 3-colourable, respectively. In this paper, we prove that every graph in $G_{4}$ is 3-colourable. This confirms Wu, Xu and Xu's conjecture.

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