Conflict-free chromatic index of trees

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 ${\chi }_{CF}^{\prime }(G)$. Dȩbski and Przyby\a{l}o showed that $2\le {\chi }_{CF}^{\prime }(T)\le 3$ 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.