Vertex colorings without isolates

Call a vertex of a vertex-colored simple graph isolated if all its neighbors have colors other than its own. A. J. Goldman has asked: When is it possible to color b vertices of a graph black and the remaining w vertices white so that no vertex is isolated? We prove (1) if G is connected and has minimum degree 2, it is always possible unless b or w is 1; (2) if G is 2-connected, then for any pair (b, w) there is a coloring in which both monochromatic subgraphs are connected; (3) if G has vertices of degree 1, a necessary condition for a (b, w) coloring without isolates to exist is that there be a solution to a certain knapsack inequality. Next, statements generalizing (1) and (2) to n colors are presented, and current knowledge about their truth is discussed. Then various refinements of (1) and (3), more complicated to state and prove, are given. For instance, with the hypotheses of (1) at least one of the monochromatic subgraphs may be chosen to be connected. Also, the necessary knapsack inequality of (3) is, in most cases, sufficient. Throughout, some consideration is given to the algorithmic complexity of coloring (if possible) without isolates. For most graphs which might arise in practice there is an efficient algorithm for the 2-color problem. However, for arbitrary graphs the 2-(or more) color problem is NP-complete.