Conflict-free coloring on open neighborhoods of claw-free graphs
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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 χ_CF^ON(G) (χ_CF^CN(G)). Dębski and Przybyło in [J. Graph Theory, 2021] showed that if G is a line graph with maximum degree Δ, then χ_CF^CN(G) = O(lnΔ). 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≥ 3, we show that if G is K_1,k-free, then χ_CF^ON(G) = O(k^2lnΔ). 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.
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