On star edge colorings of bipartite and subcubic graphs

TL;DR

This paper establishes upper bounds for the star chromatic index of bipartite graphs, including tight bounds for small parts, and confirms the conjecture for cubic Halin graphs, advancing understanding of star edge colorings.

Contribution

It provides new upper bounds for star chromatic index in bipartite graphs, including tight bounds for small parts and confirms the conjecture for cubic Halin graphs.

Findings

Tight upper bounds for complete bipartite graphs with small parts.

Sharp bounds for bipartite graphs with degree constraints.

Confirmed the star chromatic index conjecture for cubic Halin graphs.

Abstract

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 complete bipartite graphs; in particular we obtain tight upper bounds for the case when one part has size at most $3$. We also consider 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 ($k\geq 1$), we prove that if $b=2k+1$ then $\chi'_{st}(G) \leq 3k+2$; and if $b=2k$, then $\chi'_{st}(G) \leq 3k$; both upper bounds are sharp. 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.