Cliques in Graphs Excluding a Complete Graph Minor

TL;DR

This paper determines the maximum number of k-cliques in n-vertex graphs that exclude a complete graph minor, providing exact results for t ≤ 9 and all k, advancing extremal graph theory.

Contribution

It provides exact maximum counts of k-cliques in graphs excluding a K_t-minor for t ≤ 9, extending extremal results to higher clique counts.

Findings

Exact maximum number of k-cliques for t ≤ 9

Maximum total number of cliques for t ≤ 9

Observations on general t cases

Abstract

This paper considers the following question: What is the maximum number of $k$-cliques in an $n$-vertex graph with no $K_t$-minor? This question generalises the extremal function for $K_t$-minors, which corresponds to the $k=2$ case. The exact answer is given for $t\leq 9$ and all values of $k$. We also determine the maximum total number of cliques in an $n$-vertex graph with no $K_t$-minor for $t\leq 9$. Several observations are made about the case of general $t$.