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

The coloring problem (i.e., computing the chromatic number of a graph) can be solved in O^*(2n) 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(1.33n) time (Beigel and Eppstein, 2005) and 4-coloring can be solved in O(1.73n) time (Fomin, Gaspers and Saurabh, 2007). Surprisingly, for k>4 no improvements over the general O^*(2n) are known. We show that both 5-coloring and 6-coloring can also be solved in Oleft(left(2-varepsilonright)nright) 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,α>0, the chromatic number of graphs with at least α· n vertices of degree at most Delta can be computed in Oleft(left(2-varepsilonright)nright) time, for some varepsilon = varepsilonDelta,α > 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.