Distributed coloring of graphs with an optimal number of colors

This paper studies sufficient conditions to obtain efficient distributed algorithms coloring graphs optimally (i.e.\ with the minimum number of colors) in the LOCAL model of computation. Most of the work on distributed vertex coloring so far has focused on coloring graphs of maximum degree $\Delta$ with at most $\Delta+1$ colors (or $\Delta$ colors when some simple obstructions are forbidden). When $\Delta$ is sufficiently large and $c\ge \Delta-k_\Delta+1$, for some integer $k_\Delta\approx \sqrt{\Delta}-2$, we give a distributed algorithm that given a $c$-colorable graph $G$ of maximum degree $\Delta$, finds a $c$-coloring of $G$ in $\min\{O((\log\Delta)^{1/12}\log n), 2^{O(\log \Delta+\sqrt{\log \log n})}\}$ rounds, with high probability. The lower bound $\Delta-k_\Delta+1$ is best possible in the sense that for infinitely many values of $\Delta$, we prove that when $\chi(G)\le \Delta -k_\Delta$, finding an optimal coloring of $G$ requires $\Omega(n)$ rounds. Our proof is a light adaptation of a remarkable result of Molloy and Reed, who proved that for $\Delta$ large enough, for any $c\ge \Delta - k_\Delta$ deciding whether $\chi(G)\le c$ is in {\textsf{P}}, while Embden-Weinert \emph{et al.}\ proved that for $c\le \Delta-k_\Delta-1$, the same problem is {\textsf{NP}}-complete. Note that the sequential and distributed thresholds differ by one. We also show that for any sufficiently large $\Delta$, and $\Omega(\log \Delta)\le k \le \Delta/100$, every graph of maximum degree $\Delta$ and clique number at most $\Delta-k$ can be efficiently colored with at most $\Delta-\varepsilon k$ colors, for some absolute constant $\varepsilon >0$, with a randomized algorithm running in $O(\log n/\log \log n)$ rounds with high probability.