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.

翻译：彩色问题( 即计算图表的色数) 可以用美元( ) (2. 3美元) 解决, 如 Bj\ " orklund, Husfeldt 和 Koivisto 所显示的 2009 年, $k= 34美元, 以美元颜色问题为已知的更好的算法。 $( 1.33 美元) 时间( Beigel 和 Eppstein, 2005) 和 4美元 彩色问题可以用美元时间解决 $( 1. 73 美元) 解决( Fom、 Gaspers 和 Saurabrahbah, 2007 令人惊讶的是, $> 美元= 美元= 美元= 3, 美元= 34美元。 我们显示, $( left) ( left ( 2calepslon)) 时间可以解决 3美元 颜色问题( ) 和 美元( right) 时间可以解决一些 $( varepluslevlation>0) 。 作为关键步骤, 我们在计算 $ dalexal deal dealational= dalal dal date leas leas a exal exal exal exal exal ex, ex exal lex lex legal lex lex lex lex lex lex lex lex 。