On star edge colorings of bipartite and subcubic graphs

A star edge coloring of a graph is a proper edge coloring with no 2-colored path or cycle of length four. The star chromatic index chi(st)(G) of G is the minimum number t for which G has a star edge coloring with t colors. We prove upper bounds for the star chromatic index of bipartite graphs G where all vertices in one part have maximum degree 2 and all vertices in the other part has maximum degree b. Let k be an integer (k >= 1); we prove that if b = 2k + 1, then chi(st)(G) <= 3k + 2; and if b = 2k, then chi(st)(G) < 3k; both upper bounds are sharp. We also consider complete bipartite graphs; in particular we determine the star chromatic index of such graphs when one part has size at most 3, and prove upper bounds for the general case. Finally, we consider the well-known conjecture that subcubic graphs have star chromatic index at most 6; in particular we settle this conjecture for cubic Halin graphs. (C) 2021 The Authors. Published by Elsevier B.V.Funding Agencies|Swedish Research CouncilSwedish Research CouncilEuropean Commission [2017-05077]; French ANRFrench National Research Agency (ANR) [ANR-17-CE40-0022]