The maximum number of cliques in graphs without long cycles

TL;DR

This paper generalizes classical extremal graph results to bound the number of s-cliques in graphs without long cycles, identifying extremal structures and extending to graphs with no long paths.

Contribution

It extends Kopylov's theorem to bound s-cliques in graphs with limited cycle length and characterizes extremal graphs maximizing clique counts.

Findings

Extends Kopylov's theorem to s-cliques

Identifies extremal graphs for clique counts

Provides extremal number of s-cliques in graphs without long paths

Abstract

The Erd\H{o}s--Gallai Theorem states that for $k\geq 3$ every graph on $n$ vertices with more than $\frac{1}{2}(k-1)(n-1)$ edges contains a cycle of length at least $k$. Kopylov proved a strengthening of this result for 2-connected graphs with extremal examples $H_{n,k,t}$ and $H_{n,k,2}$. In this note, we generalize the result of Kopylov to bound the number of $s$-cliques in a graph with circumference less than $k$. Furthermore, we show that the same extremal examples that maximize the number of edges also maximize the number of cliques of any fixed size. Finally, we obtain the extremal number of $s$-cliques in a graph with no path on $k$-vertices.