Strong edge coloring of planar graphs with girth at least six

An edge coloring is a strong edge coloring if each path of length three uses three distinct colors. The strong chromatic index χ′s(G) of a graph G is the minimum number of colors needed in a strong edge coloring of G. In this paper, we prove that if G is a graph with ∆(G) ≤ 4 and maximum average degree less than 3, then χ′s(G) ≤ 3∆(G) + 1. Moreover, we prove that every planar graph G with girth at least 6 admits a strong edge coloring with 3∆(G) + 1 colors, which improves a result in [D. Hudák, B. Lužar, R. Soták and R. Škrekovski, Strong edge-coloring of planar graphs, Discrete Math. 324 (2014) 41–49]. In addition, we prove that if G is a planar graph with maximum degree at least 4 and girth at least 7, then χ′s(G) ≤ 3∆(G). Especially, all the results can be applied to the list version of strong edge coloring. 1