Further results on the number of cliques in graphs covered by long cycles

TL;DR

This paper characterizes the extremal graphs that maximize the number of s-cliques within a specific class of 2-connected graphs containing edges not on long cycles, extending recent results on clique counts.

Contribution

It provides a complete characterization of extremal graphs maximizing s-cliques in the class amma(n,k), building on recent work that determined the maximum number.

Findings

Characterization of extremal graphs in amma(n,k)

Extension of previous maximum clique results

Identification of structural properties of extremal graphs

Abstract

Let $\Gamma(n,k)$ be the set of $2$-connected $n$-vertex graphs containing an edge that is not on any cycle of length at least $k+1.$ Let $g_s(n,k)$ denote the maximum number of $s$-cliques in a graph in $\Gamma(n,k).$ Recently, Ji and Ye [SIAM J. Discrete Math., 37 (2023) 917-924] determined $g_s(n,k).$ They remark that it is interesting to characterize the extremal graphs. In this paper, we give such a characterization.