Backbone colorings for networks

Abstract

We study backbone colorings, a variation on classical vertex colorings: Given a graph G = (V, E) and a spanning subgraph H (the backbone) of G, a backbone coloring for G and H is a proper vertex coloring V → {1,2,...} in which the colors assigned to adjacent vertices in H differ by at least two. We concentrate on the cases where the backbone is either a spanning tree or a spanning path. 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 χ(G); for path backbones this factor is roughly 3/2. In the special case of split graphs G, the difference from χ(G) is at most an additive constant 2 or 1, for tree backbones and path backbones, respectively. The computational complexity of the problem 'Given a graph G, a spanning tree T of G, and an integer ℓ, 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). © Springer-Verlag Berlin Heidelberg 2003.