The strong clique index of a graph with forbidden cycles

Abstract

AbstractGiven a graph , the strong clique index of , denoted , is the maximum size of a set of edges such that every pair of edges in has distance at most 2 in the line graph of . As a relaxation of the renowned Erdős–Nešetřil 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 is a ‐free graph with , then , and if is a ‐free bipartite graph, then . 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 ‐free graph with satisfies , when either or and is also ‐free. Regarding the first conjecture, we prove an upper bound that is off by the constant term. Namely, for , we prove that a ‐free graph with satisfies . This improves some results of Cames van Batenburg, Kang, and Pirot.