Abstract
AbstractWe 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.
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