Extremal number of cliques of given orders in graphs with a forbidden clique minor

TL;DR

This paper establishes near-optimal bounds on the maximum number of cliques of a given size in graphs that exclude a certain clique minor, advancing understanding of extremal graph structures with forbidden minors.

Contribution

It determines asymptotically sharp bounds on clique counts in $K_t$-minor free graphs, answering a key open question in extremal graph theory.

Findings

Bound on the number of $k$-cliques is proportional to $n C(k,t)$ with sharpness demonstrated.

The bounds are tight up to lower order terms, except when $k$ is very close to $t$.

Provides a construction of graphs achieving the maximum number of $k$-cliques.

Abstract

Alon and Shikhelman initiated the systematic study of a generalization of the extremal function. Motivated by algorithmic applications, the study of the extremal function $\text{ex}(n, K_k, K_t\text{-minor})$, i.e., the number of cliques of order $k$ in $K_t$-minor free graphs on $n$ vertices, has received much attention. In this paper, we determine essentially sharp bounds on the maximum possible number of cliques of order $k$ in a $K_t$-minor free graph on $n$ vertices. More precisely, we determine a function $C(k,t)$ such that for each $k < t$ with $t-k\gg \log_2 t$, every $K_t$-minor free graph on $n$ vertices has at most $ n C(k, t)^{1+o_t(1)}$ cliques of order $k$. We also show this bound is sharp by constructing a $K_t$-minor-free graph on $n$ vertices with $C(k, t) n$ cliques of order $k$. This bound answers a question of Wood and Fox-Wei asymptotically up to $o_t(1)$ in the exponent except the extreme values when $k$ is very close to $t$.