Conflict-free chromatic number vs conflict-free chromatic index

TL;DR

This paper investigates the maximum conflict-free chromatic number and index of graphs with bounded maximum degree, revealing that the conflict-free chromatic index in line graphs grows logarithmically with degree, which is smaller than the chromatic number's growth.

Contribution

It establishes that the extremal conflict-free chromatic index for line graphs is Θ(log Δ), significantly smaller than the known Θ(log^2 Δ) for general graphs, and extends this result to near-regular graphs.

Findings

Conflict-free chromatic index in line graphs is Θ(log Δ).

Conflict-free chromatic number in general graphs is Θ(log^2 Δ).

Results apply to near-regular graphs with minimum degree proportional to maximum degree.

Abstract

A vertex coloring of a given graph $G$ is conflict-free if the closed neighborhood of every vertex contains a unique color (i.e. a color appearing only once in the neighborhood). The minimum number of colors in such a coloring is the conflict-free chromatic number of $G$, denoted $\chi_{CF}(G)$. What is the maximum possible conflict-free chromatic number of a graph with a given maximum degree $\Delta$? Trivially, $\chi_{CF}(G)\leq \chi(G)\leq \Delta+1$, but it is far from optimal - due to results of Glebov, Szab\'o and Tardos, and of Bhyravarapu, Kalyanasundaram and Mathew, the answer in known to be $\Theta\left(\ln^2\Delta\right)$. We show that the answer to the same question in the class of line graphs is $\Theta\left(\ln\Delta\right)$ - that is, the extremal value of the conflict-free chromatic index among graphs with maximum degree $\Delta$ is much smaller than the one for conflict-free chromatic number. The same result for $\chi_{CF}(G)$ is also provided in the class of near regular graphs, i.e. graphs with minimum degree $\delta \geq \alpha \Delta$.