Tight bounds towards a conjecture of Gallai

Jun Gao; Jie Ma

Tight bounds towards a conjecture of Gallai

Combinatorics 2022-10-11 v2

Authors: Jun Gao , Jie Ma

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.

Keywords

graph theory chromatic number cliques gallai conjecture tight bound

Cite

@article{arxiv.2205.14556,
  title  = {Tight bounds towards a conjecture of Gallai},
  author = {Jun Gao and Jie Ma},
  journal= {arXiv preprint arXiv:2205.14556},
  year   = {2022}
}

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