Star 5-edge-colorings of subcubic multigraphs

TL;DR

This paper proves that subcubic multigraphs with maximum average degree less than 12/5 can be colored with five colors to avoid bi-colored paths or cycles of length four, advancing understanding of star edge-colorings.

Contribution

It establishes that all subcubic multigraphs with maximum average degree below 12/5 are star 5-edge-colorable, improving previous bounds and supporting the conjecture for this class.

Findings

Subcubic multigraphs with max average degree < 12/5 are star 5-edge-colorable.

Improves previous bounds from 14/5 to 12/5 for star 5-edge-colorability.

Supports the conjecture that subcubic multigraphs are star 6-edge-colorable.

Abstract

The star chromatic index of a multigraph $G$, denoted $\chi'_{s}(G)$, 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 $\chi'_{s}(G)\le k$. Dvo\v{r}\'ak, Mohar and \v{S}\'amal [Star chromatic index, J Graph Theory 72 (2013), 313--326] proved that every subcubic multigraph is star $7$-edge-colorable, and conjectured that every subcubic multigraph should be star $6$-edge-colorable. Kerdjoudj, Kostochka and Raspaud considered the list version of this problem for simple graphs and proved that every subcubic graph with maximum average degree less than $7/3$ is star list-$5$-edge-colorable. It is known that a graph with maximum average degree $14/5$ is not necessarily star $5$-edge-colorable. In this paper, we prove that every subcubic multigraph with maximum average degree less than $12/5$ is star $5$-edge-colorable.