Breaking the $2^n$ barrier for 5-coloring and 6-coloring

TL;DR

This paper presents new algorithms that solve 5-coloring and 6-coloring problems faster than the previous $2^n$ time barrier, and extends these improvements to a broad class of graphs and list coloring.

Contribution

The authors develop algorithms that break the $2^n$ barrier for 5- and 6-coloring, and generalize exponential improvements to graphs with certain degree constraints and list coloring.

Findings

5- and 6-coloring can be solved in $O((2- ext{epsilon})^n)$ time

Chromatic number computation is faster for graphs with many low-degree vertices

Improvements extend to list coloring for the first time

Abstract

The coloring problem (i.e., computing the chromatic number of a graph) can be solved in $O^*(2^n)$ time, as shown by Bj\"orklund, Husfeldt and Koivisto in 2009. For $k=3,4$, better algorithms are known for the $k$-coloring problem. $3$-coloring can be solved in $O(1.33^n)$ time (Beigel and Eppstein, 2005) and $4$-coloring can be solved in $O(1.73^n)$ time (Fomin, Gaspers and Saurabh, 2007). Surprisingly, for $k>4$ no improvements over the general $O^*(2^n)$ are known. We show that both $5$-coloring and $6$-coloring can also be solved in $O\left(\left(2-\varepsilon\right)^n\right)$ time for some $\varepsilon>0$. As a crucial step, we obtain an exponential improvement for computing the chromatic number of a very large family of graphs. In particular, for any constants $\Delta,\alpha>0$, the chromatic number of graphs with at least $\alpha\cdot n$ vertices of degree at most $\Delta$ can be computed in $O\left(\left(2-\varepsilon\right)^n\right)$ time, for some $\varepsilon = \varepsilon_{\Delta,\alpha} > 0$. This statement generalizes previous results for bounded-degree graphs (Bj\"orklund, Husfeldt, Kaski, and Koivisto, 2010) and graphs with bounded average degree (Golovnev, Kulikov and Mihajilin, 2016). We generalize the aforementioned statement to List Coloring, for which no previous improvements are known even for the case bounded-degree graphs.