Conflict-free coloring on open neighborhoods of claw-free graphs
(2112.12173)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 Przyby\l{}o 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 $k\geq 3$, we show that if $G$ is $K{1,k}$-free, then $\chi{CF}{ON}(G) = O(k2\ln \Delta)$. 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 D\k{e}bski and Przyby\l{}o.
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