Backbone colorings for graphs: Tree and path backbones

We introduce and study backbone colorings, a variation on classical vertex colorings: Given a graph G = (V, E) and a spanning subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex coloring V -> {1, 2,...} of G in which the colors assigned to adjacent vertices in H differ by at least two. We study the cases where the backbone is either a spanning tree or a spanning path. We show that for tree backbones of G the number of colors needed for a backbone coloring of G can roughly differ by a multiplicative factor of at most 2 from the chromatic number X(G); for path backbones this factor is roughly (3)/(2). We show that the computational complexity of the problem "Given a graph G, a spanning tree T of G, and an integer e, is there a backbone coloring for G and T with at most l colors?" jumps from polynomial to NP-complete between l = 4 (easy for all spanning trees) and l = 5 (difficult even for spanning paths). We finish the paper by discussing some open problems. (c) 2007 Wiley Periodicals, Inc.