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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. 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Kang, Wouter Cames van Batenburg","submitted_at":"2019-03-14T15:45:20Z","abstract_excerpt":"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. 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