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.
Highlights
A proper vertex of a graph G is an assignment of colors to the vertices of G such that no two adjacent vertices receive the same color
Liu and Deng [10] presented an upper bound on the star chromatic index of graphs with maximum degree ∆ 7
We present a polynomial time algorithm that determines the star chromatic index of 2H-trees by finding an optimum star edge coloring of them
Summary
A proper vertex (edge coloring) of a graph G is an assignment of colors to the vertices (edges) of G such that no two adjacent vertices (edges) receive the same color. By a polynomial time algorithm, we determine the star chromatic index of every tree For this purpose, we first define a Havel-Hakimi type problem. We show that this Havel-Hakimi type problem is polynomially equivalent to the problem of existence of a star edge coloring of a tree with diameter at most four (or a 2H-tree for short), with specific number of colors Using this equivalency, we present a polynomial time algorithm that determines the star chromatic index of 2H-trees by finding an optimum star edge coloring of them. Using these bounds we find a formula for the star chromatic index of certain 2H-trees and the caterpillars (a caterpillar is a tree for which removing the leaves produces a path)
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