Tight bounds towards a conjecture of Gallai

TL;DR

This paper establishes a tight upper bound on the number of (k-1)-cliques in certain minimal graphs with chromatic number k, confirming a conjecture of Gallai and solving a problem posed by Abbott and Zhou.

Contribution

It proves a precise bound on clique counts in minimal chromatic graphs, advancing understanding of Gallai's conjecture and resolving an open problem.

Findings

Proves that such graphs contain at most n-k+3 (k-1)-cliques.

Provides a tight bound confirming Gallai's conjecture.

Answers an open problem by Abbott and Zhou.

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.