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arXiv · 2205.14556

Tight bounds towards a conjecture of Gallai

Jun Gao·Jie Ma
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Abstract

We prove that for $n>k\geq 3$, if $G$ is an $n$-vertex graph with chromatic number $k$ but any its proper subgraph has smaller chromatic number, then $G$ contains at most $n-k+3$ copies of cliques of size $k-1$. This answers a problem of Abbott and Zhou and provides a tight bound on a conjecture of Gallai.

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Jun Gao, Jie Ma. 2022-10-10. Tight bounds towards a conjecture of Gallai. https://arxiv.org/abs/2205.14556

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