Conflict-free coloring on open neighborhoods of claw-free graphs

Abstract

The `Conflict-Free Open (Closed) Neighborhood coloring', abbreviated CFON (CFCN) coloring, of a graph $G$ using $r$ colors is a coloring of the vertices of $G$ such that every vertex sees some color exactly once in its open (closed) neighborhood. The minimum $r$ such that $G$ has a CFON (CFCN) coloring using $r$ colors is called the `CFON chromatic number' (`CFCN chromatic number') of $G$. This is denoted by ${\chi }_{CF}^{ON}(G)$ (${\chi }_{CF}^{CN}(G)$). D\k ebski and Przybyło in [J. Graph Theory, 2021] showed that if $G$ is a line graph with maximum degree $\Delta$, then ${\chi }_{CF}^{CN}(G)=O(\ln \Delta )$. As an open question, they asked if the result could be extended to claw-free ($K_{1,3}$-free) graphs, which are a superclass of line graphs. For $k\ge 3$, we show that if $G$ is $K_{1,k}$-free, then ${\chi }_{CF}^{ON}(G)=O(k^2\ln \Delta )$. Since it is known that the CFCN chromatic number of a graph is at most twice its CFON chromatic number, this answers the question posed by Dębski and Przybyło.

6 pages