Λ-backbone colorings along pairwise disjoint stars and matchings.

Abstract

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.