The strong clique number of graphs with forbidden cycles
Abstract
Given a graph G, the strong clique number of G, denoted ωS(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 Erdos--Nesetril 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 C2k-free graph, then ωS(G)≤ (2k-1)(G)-2k-1 2, and if G is a C2k-free bipartite graph, then ωS(G)≤ k(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 \C5, C2k\-free graph G with (G) 1 satisfies ωS(G)≤ k(G)-(k-1), when either k≥ 4 or k∈ \2,3\ and G is also C3-free. Regarding the first conjecture, we prove an upper bound that is off by the constant term. Namely, for k≥ 3, we prove that a C2k-free graph G with (G) 1 satisfies ωS(G)≤ (2k-1)(G)+(2k-1)2. This improves some results of Cames van Batenburg, Kang, and Pirot.