A Polynomial Time Algorithm to Find the Star Chromatic Index of Trees

TL;DR

This paper introduces a polynomial time algorithm for optimally star edge coloring trees, establishes tight bounds for trees with small diameters, and derives formulas for specific tree families, advancing understanding of star chromatic indices.

Contribution

It presents the first polynomial time algorithm for optimal star edge coloring of trees and provides bounds and formulas for particular tree classes.

Findings

Polynomial time algorithm for star edge coloring of trees

Tight bounds on star chromatic index for trees with diameter ≤ 4

Formulas for star chromatic index of certain tree families

Abstract

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every path and cycle of length four in $G$ uses at least three different colors. The star chromatic index of a graph $G$, is the smallest integer $k$ for which $G$ admits a star edge coloring with $k$ colors. In this paper, we present a polynomial time algorithm that finds an optimum star edge coloring for every tree. We also provide some tight bounds on the star chromatic index of trees with diameter at most four, and using these bounds we find a formula for the star chromatic index of certain families of trees.