Before revision:  In particular, for any constants k, \Delta,\alpha>0, k-coloring for graphs with at least \alpha\cdot n vertices of degree at most \Delta can be solved in O\left(\left(2-\varepsilon\right)^n\right) time, for some \varepsilon = k, \Delta,\alpha As a consequence, for any constant k we can solve k-coloring exponentially faster than O^*(2^n) for sparse graphs.
After revision:  In particular, for any constants k, \Delta,\alpha>0, k-coloring for graphs with at least \alpha\cdot n vertices of degree at most \Delta can be solved in O\left(\left(2-\varepsilon\right)^n\right) time, for some \varepsilon = k, \Delta,\alpha 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.