λ-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 number of colors needed for a λ-backbone coloring of (G,S) 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 `, is there a λ-backbone coloring of (G, S) with at most ` colors?” jumps from polynomially solvable to NP-complete between ` = λ+1 and ` = λ + 2 (difficult even for matchings). We finish the paper by discussing some open problems regarding planar graphs.