Asymptotically Tight Bound for the Conflict-Free Chromatic Index

Abstract

The conflict-free chromatic index of a graph $G$ is the minimum number of colours in an edge colouring of $G$ such that the neighbourhood of every edge contains a colour appearing exactly once. Its vertex analogue is the conflict-free chromatic number. These two parameters naturally coincide when the second is applied to the line graph of $G$. It is known that two variants of the latter parameter exhibit substantially different behaviour. For closed vertex neighbourhoods, where each vertex belongs to its own neighbourhood, it is known that $O(\ln^2 \Delta)$ colours suffice, where $\Delta$ denotes the maximum degree of $G$, and this bound is tight in order. In contrast, for open neighbourhoods, the corresponding parameter can be as large as $\Delta+1$, but is bounded above by $O(\ln^{2+\varepsilon} \Delta)$ for claw-free graphs. Since line graphs are claw-free, this yields the best known general upper bound for the edge analogue in the setting of open neighbourhoods. For closed edge neighbourhoods, a stronger general upper bound of $3\log_2 \Delta + 4$ is known. In this paper, we show that for both variants, the conflict-free chromatic index is bounded above by $(1+o(1))\log_2 \Delta$. Since complete graphs require at least $(1 - o(1)) \log_2 \Delta$ colours in the closed as well as the open setting, our result is asymptotically tight in order and in the leading constant. Moreover, we strengthen this conclusion by showing that $(1 - o(1)) \log_2 \Delta$ colours are also typically necessary, as we prove this asymptotically almost surely for random graphs in both dense and relatively sparse regimes. Our proofs combine the probabilistic method with deterministic graph decomposition techniques, as well as new results relating the parameters under consideration with the chromatic number of a graph.