Improved Upper Bounds for λ-Backbone Colorings Along Matchings and Stars

Abstract

We continue the study on backbone colorings, a variation on classical vertex colorings that was introduced at WG2003. Given a graph G= (V,E) and a spanning subgraph Hof G(the backbone of G), a i¾?-backbone coloring for Gand His a proper vertex coloring Vi¾?{1,2,...} of Gin which the colors assigned to adjacent vertices in Hdiffer by at least i¾?. The main outcome of earlier studies is that the minimum number i¾? of colors for which such colorings Vi¾?{1,2,...,i¾?} exist in the worst case is a factor times the chromatic number (for all studied types of backbones). We show here that for split graphs and matching or star backbones, i¾? is at most a small additive constant (depending on i¾?) higher than the chromatic number. Despite the fact that split graphs have a nice structure, these results are difficult to prove. Our proofs combine algorithmic and combinatorial arguments. We also indicate other graph classes for which our results imply better upper bounds on i¾? than the previously known bounds.