Conflict-free chromatic number vs conflict-free chromatic index

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 $χ_{CF}(G)$. What is the maximum possible conflict-free chromatic number of a graph with a given maximum degree $Δ$? Trivially, $χ_{CF}(G)\leq χ(G)\leq Δ+1$, but it is far from optimal - due to results of Glebov, Szabó and Tardos, and of Bhyravarapu, Kalyanasundaram and Mathew, the answer in known to be $Θ\left(\ln^2Δ\right)$. We show that the answer to the same question in the class of line graphs is $Θ\left(\lnΔ\right)$ - that is, the extremal value of the conflict-free chromatic index among graphs with maximum degree $Δ$ is much smaller than the one for conflict-free chromatic number. The same result for $χ_{CF}(G)$ is also provided in the class of near regular graphs, i.e. graphs with minimum degree $δ\geq αΔ$.