A note on a conjecture of Wu, Xu and Xu

摘要

Let l >= 2 be a positive integer and let gl denote the family of graphs which have girth 2l + 1 and have no holes of odd length at least 2l + 3. Chudnovsky and Seymour proved that every graph G is an element of g(2) is three-colorable. In 2022, Wu, Xu and Xu conjectured that every graph G is an element of boolean OR 1 gl is three-colorable. Soon after, Wu, Xu and Xu confirmed the case l = 3. In this note, we prove that every graph G is an element of gl with radius at most l + 1 is three-colorable. This generalizes the results of Xu, Yu and Zha in 2017.

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