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

TL;DR

This paper extends conflict-free coloring bounds from line graphs to broader classes of claw-free graphs, showing that the open neighborhood conflict-free chromatic number grows logarithmically with maximum degree for these classes.

Contribution

It proves that for any $K_{1,k}$-free graph, the conflict-free open neighborhood chromatic number is bounded by a function of $k^2$ times the logarithm of maximum degree, extending previous results.

Findings

Bound $oldsymbol{ ext{CFON}}$ chromatic number for $K_{1,k}$-free graphs.

Shows $oldsymbol{ ext{CFCN}}$ number is at most twice $oldsymbol{ ext{CFON}}$ number.

Answers an open question by extending bounds to 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 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 $\chi_{CF}^{ON}(G) = O(k^2\ln \Delta)$. 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 D\k{e}bski and Przyby\l{}o.