Conflict-free chromatic index of trees

TL;DR

This paper introduces an efficient linear-time algorithm to determine the conflict-free chromatic index of certain trees, advancing understanding of conflict-free edge colorings in graph theory.

Contribution

It provides a linear-time algorithm for calculating the conflict-free chromatic index of trees without 2-degree vertices, addressing an open question.

Findings

The conflict-free chromatic index of such trees is either 2 or 3.

The algorithm operates in linear time, O(|V(T)|).

This work partially answers an open question in the field.

Abstract

A graph $G$ is conflict-free $k$-edge-colorable if there exists an assignment of $k$ colors to $E(G)$ such that for every edge $e\in E(G)$, there is a color that is assigned to exactly one edge among the closed neighborhood of $e$. The smallest $k$ such that $G$ is conflict-free $k$-edge-colorable is called the conflict-free chromatic index of $G$, denoted $\chi'_{CF}(G)$. D\c{e}bski and Przyby\a{l}o showed that $2\le\chi'_{CF}(T)\le 3$ for every tree $T$ of size at least two. In this paper, we present an algorithm to determine the conflict-free chromatic index of a tree without 2-degree vertices, in time $O(|V(T)|)$. This partially answer a question raised by Kamyczura, Meszka and Przyby\a{l}o.