On the number of cliques in graphs with a forbidden subdivision or immersion

TL;DR

This paper establishes upper bounds on the number of cliques in graphs with forbidden subdivisions or immersions, advancing extremal graph theory by improving known bounds and proposing conjectures for optimal limits.

Contribution

It provides new upper bounds for the maximum number of cliques in graphs excluding a fixed clique subdivision or immersion, improving previous results and suggesting conjectures for the optimal bounds.

Findings

Graphs with no $K_t$-immersion have at most $2^{t+ ext{log}^2 t}n$ cliques.

Graphs with no $K_t$-subdivision have at most $2^{1.817t}n$ cliques.

The bounds are sharp up to a factor of $2^{O( ext{log}^2 t)}$.

Abstract

How many cliques can a graph on $n$ vertices have with a forbidden substructure? Extremal problems of this sort have been studied for a long time. This paper studies the maximum possible number of cliques in a graph on $n$ vertices with a forbidden clique subdivision or immersion. We prove for $t$ sufficiently large that every graph on $n \geq t$ vertices with no $K_t$-immersion has at most $2^{t+\log^2 t}n$ cliques, which is sharp apart from the $2^{O(\log^2 t)}$ factor. We also prove that the maximum number of cliques in an $n$-vertex graph with no $K_t$-subdivision is at most $2^{1.817t}n$. This improves on the best known exponential constant by Lee and Oum. We conjecture that the optimal bound is $3^{2t/3 +o(t)}n$, as we proved for minors in place of subdivision in earlier work.