Extending Graph Colorings

Suppose χ(G)=r and P⊆V(G). It is known that if the distance between any two vertices in P is at least 4, then any (r+1)-coloring of P extends to an (r+1)-coloring of all of G, but an r-coloring of P might not extend to an r-coloring of G. We show that if the distance between any two vertices in P is at least 3, then an (r+1)-coloring of P can be extended to a ⌈(3r+1)/2⌉-coloring of G. Kostochka showed that if P induces a set of k-cliques whose pairwise distance is at least 4k, then an (r+1)-coloring of P can be extended to an (r+1)-coloring of G. We give Kostochkaʹs proof and more precise results concerning the distance required between precolored components. For example, we show that when k=r, there is a coloring extension provided the cliques have pairwise distance at least 3k. We relate the structure of the precolored components to the number of extra colors needed in a coloring extension theorem. We construct families of graphs to show that all of the above results are close to being best possible.