Star chromatic index of subcubic multigraphs

• Published in: Journal of graph theory Vol. 88; no. 4; pp. 566 - 576
• Main Authors: Lei, Hui, Shi, Yongtang, Song, Zi‐Xia
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc 01.08.2018
• Subjects: Graph coloring; Graph theory; maximum average degree; star edge‐coloring; subcubic multigraphs
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.22230

The star chromatic index of a multigraph G, denoted χs′(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 bicolored. A multigraph G is star k‐edge‐colorable if χs′(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 article that every subcubic multigraph should be star 6‐edge‐colorable. In this article, we first prove that it is NP‐complete to determine whether χs′(G)≤3 for an arbitrary graph G. This answers a question of Mohar. We then establish some structure results on subcubic multigraphs G with δ(G)≤2 such that χs′(G)>k but χs′(G−v)≤k for any v∈V(G), 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.