Precoloring Extensions of Brooks' Theorem

Let G be a connected graph with maximum degree k (other than a complete graph or odd cycle), let W be a precolored set of vertices in G inducing a subgraph F , and let D be the minimum distance in G between components of F . If the components of F are complete graphs and D ≥ 8 (for k ≥ 4) or D ≥ 10 (for k = 3), then every proper k-coloring of F extends to a proper k-coloring of G. If the components of F are single vertices and D ≥ 8, and the vertices outside W are assigned color lists of size k, then every k-coloring of F extends to a proper coloring of G with the color on each vertex chosen from its list. These results are sharp.