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arXiv · 2409.10899

Conflict-free chromatic index of trees

Shanshan Guo·Ethan Y. H. Li·Luyi Li·Ping Li
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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.

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Shanshan Guo, Ethan Y. H. Li, Luyi Li, Ping Li. 2024-09-17. Conflict-free chromatic index of trees. https://arxiv.org/abs/2409.10899

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