Strong cliques and forbidden cycles

Given a graph $G$, the strong clique number ${\omega }_2^{\mathrm{\prime }}(G)$ 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 $\mathrm{\Delta }$, we show among other results the following: ${\omega }_2^{\mathrm{\prime }}(G)\le 5{\mathrm{\Delta }}^2\mathrm{/}4$ if $G$ is triangle-free; ${\omega }_2^{\mathrm{\prime }}(G)\le 3(\mathrm{\Delta }-1)$ if $G$ is $C_4$-free; ${\omega }_2^{\mathrm{\prime }}(G)\le {\mathrm{\Delta }}^2$ 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.