The number of cliques in graphs covered by long cycles

TL;DR

This paper establishes an upper bound on the number of s-cliques in 2-connected graphs with certain cycle length restrictions, confirming a conjecture and linking clique counts to cycle coverage.

Contribution

It proves a conjecture by Ma and Yuan, providing a clique count bound based on cycle length constraints and characterizing edges covered by long cycles.

Findings

Proves an upper bound on N_s(G) for graphs with edges not on long cycles.

Confirms a conjecture of Ma and Yuan regarding clique counts.

Shows that exceeding the bound implies every edge lies on a long cycle.

Abstract

Let $G$ be a 2-connected $n$-vertex graph and $N_s(G)$ be the total number of $s$-cliques in $G$. Let $k\ge 4$ and $s\ge 2$ be integers. In this paper, we show that if $G$ has an edge $e$ which is not on any cycle of length at least $k$, then $N_s(G)\le r{k-1\choose s}+{t+2\choose s}$, where $n-2=r(k-3)+t$ and $0\le t\le k-4$. This result settles a conjecture of Ma and Yuan and provides a clique version of a theorem of Fan, Wang and Lv. As a direct corollary, if $N_s(G)> r{k-1\choose s}+{t+2\choose s}$, every edge of $G$ is covered by a cycle of length at least $k$.