# Strong cliques and forbidden cycles

## TL;DR

This paper investigates the maximum size of strong cliques in graphs that lack certain cycle lengths, providing new bounds that extend previous research and are achieved by natural extremal examples.

## Contribution

It introduces improved bounds on the strong clique number for graphs missing specific cycle lengths, extending prior work in the field.

## Key findings

- Bound of 5Δ²/4 for triangle-free graphs
- Bound of 3(Δ-1) for C4-free graphs
- Bound of Δ² for graphs missing odd cycles

## Abstract

Given a graph $G$, the strong clique number $\omega_2'(G)$ of $G$ is the cardinality of a largest collection of edges every pair of which are incident or connected by an edge in $G$. We study the strong clique number of graphs missing some set of cycle lengths. For a graph $G$ of large enough maximum degree $\Delta$, we show among other results the following: $\omega_2'(G)\le5\Delta^2/4$ if $G$ is triangle-free; $\omega_2'(G)\le3(\Delta-1)$ if $G$ is $C_4$-free; $\omega_2'(G)\le\Delta^2$ if $G$ is $C_{2k+1}$-free for some $k\ge 2$. These bounds are attained by natural extremal examples. Our work extends and improves upon previous work of Faudree, Gy\'arf\'as, Schelp and Tuza (1990), Mahdian (2000) and Faron and Postle (2019). We are motivated by the corresponding problems for the strong chromatic index.