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

  • Authors:

  • Hajo Broersma;Bert Marchal;Daniel Paulusma;A. N. Salman

  • Affiliations:

  • Department of Computer Science, Durham University, DH1 3LE Durham, United Kingdom;Faculty of Economics and Business Administration, Department of Quantitative Economics, University of Maastricht, PO Box 616, 6200 MD Maastricht, The Netherlands;Department of Computer Science, Durham University, DH1 3LE Durham, United Kingdom;Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesa 10, Bandung 40132, Indonesia

  • Venue:
  • SOFSEM '07 Proceedings of the 33rd conference on Current Trends in Theory and Practice of Computer Science
  • Year:
  • 2007

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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 茂戮驴-backbone coloring for Gand His a proper vertex coloring V茂戮驴{1,2,...} of Gin which the colors assigned to adjacent vertices in Hdiffer by at least 茂戮驴. The main outcome of earlier studies is that the minimum number 茂戮驴 of colors for which such colorings V茂戮驴{1,2,...,茂戮驴} 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, 茂戮驴 is at most a small additive constant (depending on 茂戮驴) 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 茂戮驴 than the previously known bounds.