List star edge-coloring of claw-free subcubic multigraphs

Abstract

A star edge-coloring of a multigraph G is a proper edge-coloring of G such that no path or cycle of length four is bi-colored. The star chromatic index of G is the minimum number of colors needed to guarantee that G admits a star edge-coloring. The list star chromatic index of G is the smallest integer k such that for any k-uniform list assignment L for the set of edges, G has a star edge-coloring from L. Dvořák, Mohar and Šámal proved that every subcubic multigraph has star chromatic index at most 7, and conjectured that 7 can be further improved to 6. Lužar, Mockovčiaková and Soták strengthened the result of Dvořák, Mohar and Šámal by showing that every subcubic multigraph has list star chromatic index at most 7. In this paper, we verify the conjecture of Dvořák, Mohar and Šámal for a class of subcubic multigraphs. We prove that every claw-free subcubic multigraph has list star chromatic index at most 6, and give a few examples to show that the upper bound is tight.