Extremal Results on Conflict-Free Coloring

Abstract

A conflict-free open neighborhood (CFON) coloring of a graph is an assignment of colors to the vertices such that for every vertex there is a color that appears exactly once in its open neighborhood. For a graph $G$, the smallest number of colors required for such a coloring is called the CFON chromatic number and is denoted by ${\chi }_{\mathrm{ON}}\left(G\right)$. By considering closed neighborhood instead of open neighborhood, we obtain the analogous notions of conflict-free (CF) closed neighborhood (CFCN) coloring, and CFCN chromatic number (denoted by ${\chi }_{\mathrm{CN}}\left(G\right)$). The notion of CF coloring was introduced in 2002, and has since received considerable attention. We study CFON and CFCN colorings and show the following results. In what follows, $\Delta$ denotes the maximum degree of the graph.

• We show that if $G$ is a $K_{1,k}$-free graph, then ${\chi }_{\mathrm{ON}}\left(G\right)=O(k \ln \Delta )$. Dębski and Przybyło had shown that if $G$ is a line graph, then ${\chi }_{\mathrm{CN}}\left(G\right)=O(\ln \Delta )$. As an open question, they had asked if their result could be extended to claw-free ($K_{1,3}$-free) graphs, which is a superclass of line graphs. Since ${\chi }_{\mathrm{CN}}\left(G\right)\le 2{\chi }_{\mathrm{ON}}\left(G\right)$, our result answers their open question. It is known that there exist separate families of $K_{1\text{.}k}$-free graphs with ${\chi }_{\mathrm{ON}}\left(G\right)=\Omega (\ln \Delta )$ and ${\chi }_{\mathrm{ON}}\left(G\right)=\Omega \left(k\right)$.

• For a $K_{1,k}$-free graph $G$ on $n$ vertices, we show that ${\chi }_{\mathrm{CN}}\left(G\right)=O(\ln k \ln n)$. This bound is asymptotically tight for some values of $k$ since there are graphs $G$ with ${\chi }_{\mathrm{CN}}\left(G\right)=\Omega ({\ln }^2 n)$.

• Let $\delta \ge 0$ be an integer. We define $f_{\mathrm{CN}}\left(\delta \right)$ as follows:

  $f_{\mathrm{CN}}\left(\delta \right)=\max \{{\chi }_{\mathrm{CN}}\left(G\right):G\text{is a graph with minimum degree at least}\delta \}.$

  It is easy to see that $f_{\mathrm{CN}}\left({\delta }^{\prime }\right)\ge f_{\mathrm{CN}}\left(\delta \right)$ when ${\delta }^{\prime }<\delta$. Let $c$ be a positive constant. It was shown that $f_{\mathrm{CN}}\left(c\Delta \right)=\Theta (\ln \Delta )$. In this paper, we show (i) $f_{\mathrm{CN}}\left(\frac{c\Delta }{{\ln }^{ϵ} \Delta }\right)=O({\ln }^{1+ϵ} \Delta )$, where $ϵ$ is a constant such that $0\le ϵ\le 1$ and (ii) $f_{\mathrm{CN}}\left(c{\Delta }^{1-ϵ}\right)=\Omega ({\ln }^2 \Delta )$, where $ϵ$ is a constant such that $0<ϵ<0.003$. Together with the known upper bound ${\chi }_{\mathrm{CN}}\left(G\right)=O({\ln }^2 \Delta )$, this implies that $f_{\mathrm{CN}}\left(c{\Delta }^{1-ϵ}\right)=\Theta ({\ln }^2 \Delta )$.