On S-packing edge-colorings of graphs with small edge weight

The edge weight, denoted by w(e), of a graph G is max{dG(u)+dG(v):uv∈E(G)}. For an integer sequence S=(s1,s2,…,sk) with 0≤s1≤s2≤⋯≤sk, an S-packing edge-coloring of a graph G is a partition of E(G) into k subsets E1,E2,…,Ek such that for each 1≤i≤k, dL(G)(e,e′)≥si+1 for any e,e′∈Ei, where dL(G)(e,e′) denotes the distance of e and e′ in the line graph L(G) of G. Hocquard, Lajou and Lužar (Between proper and strong edge-colorings of subcubic graphs, https://arxiv.org/abs/2011.02175) posed an open problem: every subcubic bipartite graph G with w(e)≤5 is (1,24)-packing edge-colorable. We confirm the question in affirmative with a stronger way. It is shown that for any graph G (not necessarily subcubic bipartite) with w(e)≤5 is (1,24)-packing edge-colorable. We also prove that every graph G with w(e)≤6 is (1,28)-packing edge-colorable. In addition, we prove that if G is cubic graph, then it has a (1,320)-packing edge-coloring and a (1,447)-packing edge-coloring. Furthermore, if G is 3-edge-colorable, then it has a (1,318)-packing edge-coloring and a (1,442)-packing edge-coloring. These strengthen results of Gastineau and Togni (On S-packing edge-colorings of cubic graphs, Discrete Appl. Math. 259 (2019) 63–75). © 2021 Elsevier Inc.