Brooks' theorem with forbidden colors

Abstract

We consider extensions of Brooks' classic theorem on vertex coloring where some colors cannot be used on certain vertices. In particular we prove that if is a connected graph with maximum degree that is not a complete graph and is a set of vertices where either at most colors are forbidden for every vertex in , and any two vertices of are at distance at least 4, or at most colors are forbidden for every vertex in , and any two vertices of are at distance at least 3,then there is a proper ‐coloring of respecting these constraints. In fact, we shall prove that these results hold in the more general setting of list colorings. These results are sharp.