Every subcubic multigraph is (1,27) $(1,{2}^{7})$‐packing edge‐colorable

Abstract

AbstractFor a nondecreasing sequence of positive integers, an ‐packing edge‐coloring of a graph is a decomposition of edges of into disjoint sets such that for each the distance between any two distinct edges is at least . The notion of ‐packing edge‐coloring was first generalized by Gastineau and Togni from its vertex counterpart. They showed that there are subcubic graphs that are not ‐packing (abbreviated to ‐packing) edge‐colorable and asked the question whether every subcubic graph is ‐packing edge‐colorable. Very recently, Hocquard, Lajou, and Lužar showed that every subcubic graph is ‐packing edge‐colorable and every 3‐edge‐colorable subcubic graph is ‐packing edge‐colorable. Furthermore, they also conjectured that every subcubic graph is ‐packing edge‐colorable.In this paper, we confirm the conjecture of Hocquard, Lajou, and Lužar, and extend it to multigraphs.