Star chromatic index of subcubic multigraphs

Abstract

The star chromatic index of a multigraph G, denoted χ ′ s (G), is the minimum number of colors needed to properly color the edges of G such that no path or cycle of length four is bi-colored. A multigraph G is star k-edge-colorable if χ ′ s (G) ≤ k. Dvořák, Mohar and Šámal [Star chromatic index, J. Graph Theory 72 (2013), 313-326] proved that every subcubic multigraph is star 7-edge-colorable. They conjectured in the same paper that every subcubic multigraph should be star 6-edge-colorable. In this paper, we first prove that it is NP-complete to determine whether χ ′ s (G) ≤ 3 for an arbitrary graph G. This answers a question of Mohar. We then establish some structure results on subcubic multigraphs G with, where k ∈ {5, 6}. We finally apply the structure results, along with a simple discharging method, to prove that every subcubic multigraph G is star 6-edge-colorable if mad(G) < 5/2, and star 5-edge-colorable if mad(G) < 24/11, respectively, where mad(G) is the maximum average degree of a multigraph G. This partially confirms the conjecture of Dvořák, Mohar and Šámal.

Star 5-edge-colorings of subcubic multigraphs

“…Since every multigraph with maximum degree at most two or number of vertices at most four is star 5-edge-colorable, we see that ∆(G) = 3 and |G| ≥ 5. As observed in [9], any 2-vertex in G must have two distinct neighbors. The following Lemma 2.1 and Lemma 2.2 are proved in [9] and will be used in this paper.…”

List star edge coloring of generalized Halin graphs

“…But it is difficult to determine the star chromatic index of graphs even for complete graphs and subcubic graphs, just like to determine whether the chromatic index of an arbitrary graph with maximum degree ∆ is ∆ or ∆ + 1. Lei, Shi, and Song [9] proved that it is NP-complete to determine whether χ st (G) ≤ 3 for an arbitrary subcubic graph G. These motivate scholars to study some special classes of graphs.…”

On Star 5-Colorings of Sparse Graphs

“…Acyclic coloring and star coloring have been an active area of research, and we direct the readers to a thorough survey by Borodin [5] for the rich literature. There is also an edge-coloring analogue for star coloring; for recent progress on star edge-coloring subcubic graphs, see [12,15,19,20].…”