The strong clique number of graphs with forbidden cycles

TL;DR

This paper investigates the strong clique number in graphs with forbidden cycles, proving new bounds that improve previous conjectures and extend results to broader classes of graphs.

Contribution

It proves a stronger form of a conjecture on graphs without even cycles and provides improved bounds for the strong clique number in such graphs.

Findings

Proves the conjecture for graphs excluding certain cycles, including odd cycles.

Establishes upper bounds on the strong clique number for $C_{2k}$-free graphs.

Improves previous results by tightening bounds on the strong clique number.

Abstract

Given a graph $G$, the strong clique number of $G$, denoted $\omega_S(G)$, is the maximum size of a set $S$ of edges such that every pair of edges in $S$ has distance at most $2$ in the line graph of $G$. As a relaxation of the renowned Erd\H{o}s--Ne\v{s}et\v{r}il conjecture regarding the strong chromatic index, Faudree et al. suggested investigating the strong clique number, and conjectured a quadratic upper bound in terms of the maximum degree. Recently, Cames van Batenburg, Kang, and Pirot conjectured a linear upper bound in terms of the maximum degree for graphs without even cycles. Namely, if $G$ is a $C_{2k}$-free graph, then $\omega_S(G)\leq (2k-1)\Delta(G)-{2k-1\choose 2}$, and if $G$ is a $C_{2k}$-free bipartite graph, then $\omega_S(G)\leq k\Delta(G)-(k-1)$. We prove the second conjecture in a stronger form, by showing that forbidding all odd cycles is not necessary. To be precise, we show that a $\{C_5, C_{2k}\}$-free graph $G$ with $\Delta(G)\ge 1$ satisfies $\omega_S(G)\leq k\Delta(G)-(k-1)$, when either $k\geq 4$ or $k\in \{2,3\}$ and $G$ is also $C_3$-free. Regarding the first conjecture, we prove an upper bound that is off by the constant term. Namely, for $k\geq 3$, we prove that a $C_{2k}$-free graph $G$ with $\Delta(G)\ge 1$ satisfies $\omega_S(G)\leq (2k-1)\Delta(G)+(2k-1)^2$. This improves some results of Cames van Batenburg, Kang, and Pirot.