Star chromatic index

TL;DR

This paper investigates the star chromatic index of graphs, providing bounds related to maximum degree, with special focus on cubic graphs, and employs diverse mathematical techniques for proofs.

Contribution

It establishes near-linear upper bounds for the star chromatic index and characterizes graphs attaining the lower bounds, especially in cubic graphs.

Findings

Upper bound on $oldsymbol{ ext{star chromatic index}}$ in terms of maximum degree

Lower bound of $oldsymbol{ ext{star chromatic index}}$ for complete graphs

Bounds and characterizations for cubic graphs

Abstract

The star chromatic index $\chi_s'(G)$ of a graph $G$ is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored. We obtain a near-linear upper bound in terms of the maximum degree $\Delta=\Delta(G)$. Our best lower bound on $\chi_s'$ in terms of $\Delta$ is $2\Delta(1+o(1))$ valid for complete graphs. We also consider the special case of cubic graphs, for which we show that the star chromatic index lies between 4 and 7 and characterize the graphs attaining the lower bound. The proofs involve a variety of notions from other branches of mathematics and may therefore be of certain independent interest.