Strong edge-colorings of sparse graphs with large maximum degree

Abstract

A strong k-edge-coloring of a graph G is a mapping from E(G) to {1, 2, ... , k} such that every two adjacent edges or two edges adjacent to the same edge receive distinct colors. The strong chromatic index X's( G) of a graph G is the smallest integer k such that G admits a strong k-edge-coloring. We give bounds on X's(G) in terms of the maximum degree Delta(G) of a graph G when G is sparse, namely, when G is 2-degenerate or when the maximum average degree Mad(G) is small. We prove that the strong chromatic index of each 2-degenerate graph G is at most 5 Delta(G) + 1. Furthermore, we show that for a graph G, if Mad(G) < 8/3 and Delta(G) >= 9, then X's(G) <= 3 Delta(G) 3 (the bound 3 Delta(G) - 3 is sharp) and if Mad(G) < 3 and Delta(G) >= 7, then X's(G) <= 3 Delta(G) (the restriction Mad(G) < 3 is sharp). (C) 2017 Published by Elsevier Ltd.