Conflict-free chromatic index of bipartite graphs

TL;DR

This paper proves that the conflict-free chromatic index of any bipartite graph without isolated vertices is at most 3, confirming a longstanding conjecture.

Contribution

It confirms the conjecture that bipartite graphs without isolated vertices have a conflict-free chromatic index at most 3.

Findings

Confirmed the conjecture for bipartite graphs without isolated vertices.

Established an upper bound of 3 for the conflict-free chromatic index.

Provided a proof for the conjecture by Kamyczura, Meszka, and Przybyło.

Abstract

An edge coloring of a graph $G$ is called conflict-free if, for every edge, its closed neighborhood contains a color that appears exactly once. The least number of colors required for such a coloring is the conflict-free chromatic index of $G$, denoted by $\chi'_{CF}(G)$. Kamyczura, Meszka, and Przyby{\l}o conjectured that $\chi'_{CF}(G)\le 3$ for any bipartite graph $G$ without isolated vertices. In this paper, we confirm this conjecture.