Lambda-backbone Colorings along Pairwise Disjoint Stars and Matchings

Given an integer λ ≥ 2, a graph G = (V,E) and a spanning subgraph H of G (the backbone of G), a λ-backbone coloring of (G,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 λ. We study the case where the backbone is either a collection of pairwise disjoint stars or a matching. We show that for a star backbone S of G the minimum number l for which a λ-backbone coloring of (G,S) with colors in {1, . . . , l} exists can roughly differ by a multiplicative factor of at most 2 − 1 λ from the chromatic number χ(G). For the special case of matching backbones this factor is roughly 2− 2 λ+1 . We also show that the computational complexity of the problem “Given a graph G with a star backbone S, and an integer l, is there a λ-backbone coloring of (G,S) with colors in {1, . . . , l}?” jumps from polynomially solvable to NP-complete between l = λ + 1 and l = λ + 2 (the case l = λ + 2 is even NP-complete for matchings). We finish the paper by discussing some open problems regarding planar graphs.