Conflict-free chromatic index of trees

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 \(\chi'_{CF}(G)\). Dȩbski 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.