On August 30, 2022, Jun Gao (高峻) from the IBS Extremal Combinatorics and Probability Group gave a talk at the Discrete Math Seminar about the tight upper bound on the number of (k-1)-cliques in a k-critical graph, solving the conjecture of Abbott and Zhou in 1992. The title of his talk was “Number of (k-1)-cliques in k-critical graph“.
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.
This is joint work with Jie Ma.