A note on the conflict-free chromatic index

TL;DR

This paper provides a simple, explicit proof that the conflict-free chromatic index of a graph is at most three times the logarithm of its maximum degree, improving understanding of conflict-free edge colorings.

Contribution

It offers a straightforward proof that the conflict-free chromatic index is bounded by 3 log2 of the maximum degree, strengthening previous probabilistic bounds.

Findings

Conflict-free chromatic index is at most 3 log2(Δ)+1.

Bound of 4 for bipartite graphs.

Relation between conflict-free chromatic index and chromatic number.

Abstract

Let $G$ be a graph with maximum degree $\Delta$ and without isolated vertices. An edge colouring $c$ of $G$ is conflict-free if the closed neighbourhood of every edge includes a uniquely coloured element. The least number of colours admitting such $c$ is the conflict-free chromatic index of $G$, denoted by $\chi'_{CF}(G)$. In "Conflict-free chromatic number versus conflict-free chromatic index" [J. Graph Theory, 2022; 99: 349--358] it was recently proved by means of the probabilistic method that $\chi'_{CF}(G)\leq C_1\log_2\Delta+C_2$, where $C_1>337$ and $C_2$ are constants, whereas there are families of graphs with $\chi'_{CF}(G)\geq (1-o(1))\log_2\Delta$. In this note we provide an explicit simple proof of the fact that $\chi'_{CF}(G)\leq 3\log_2\Delta+1$, which is a corollary of a stronger result: $\chi'_{CF}(G)\leq 3\log_2\chi(G)+1$. For this aim we prove a few auxiliary observations, implying in particular that $\chi'_{CF}(G)\leq 4$ for bipartite graphs.