The strong clique index of a graph with forbidden cycles

Given a graph G, the strong clique index 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 Erdos-Nesetril conjecture regarding the strong chromatic index, Faudree et al. suggested investigating the strong clique index, 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 2 k-free graph with Delta ( G ) >= max { 4 , 2 k - 2 }, then omega S ( G ) <= ( 2 k - 1 ) Delta ( G ) -2 k - 1 2, and if G is a C 2 k-free bipartite graph, then omega S ( G ) <= 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 2 k }-free graph G with Delta ( G ) >= 1 satisfies omega S ( G ) <= k Delta ( G ) - ( k - 1 ), when either k >= 4 or k is an element of { 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 >= 3, we prove that a C 2 k-free graph G with Delta ( G ) >= 1 satisfies omega S ( G ) <= ( 2 k - 1 ) Delta ( G ) + ( 2 k - 1 ) 2. This improves some results of Cames van Batenburg, Kang, and Pirot.