Discussiones Mathematicae Graph Theory 29(1) (2009) 143-162

BACKBONE COLORINGS ALONG STARS AND MATCHINGS IN SPLIT GRAPHS: THEIR SPAN IS CLOSE TO THE CHROMATIC NUMBER

Hajo Broersma1, Bert Marchal2, Daniel Paulusma1 and A.N.M. Salman3

1Department of Computer Science
Durham University, Science Laboratories
South Road, Durham DH1 3LE, England
e-mail: {hajo.broersma,daniel.paulusma}@durham.ac.uk

2Faculty of Economics and Business Administration
Department of Quantitative Economics, University of Maastricht
P.O. Box 616, 6200 MD Maastricht, The Netherlands
e-mail: b.marchal@ke.unimaas.nl

3Faculty of Mathematics and Natural Sciences
Institut Teknologi Bandung
Jalan Ganesa 10, Bandung 40132, Indonesia
e-mail: msalman@math.itb.ac.id

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 H of G (the backbone of G), a λ-backbone coloring for G and 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. The algorithmic and combinatorial properties of backbone colorings have been studied for various types of backbones in a number of papers. The main outcome of earlier studies is that the minimum number l of colors, for which such colorings V→ {1,2,…,l} exist, in the worst case is a factor times the chromatic number (for path, tree, matching and star backbones). We show here that for split graphs and matching or star backbones, l is at most a small additive constant (depending on λ) higher than the chromatic number. Our proofs combine algorithmic and combinatorial arguments. We also indicate other graph classes for which our results imply better upper bounds on l than the previously known bounds.

Keywords: backbone coloring, split graph, matching, star.

2000 Mathematics Subject Classification: 05C15, 015C85, 015C17.

References

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[2] H.J. Broersma, A general framework for coloring problems: old results, new results and open problems, in: Proceedings of IJCCGGT 2003, LNCS 3330 (2005) 65-79.
[3] H.J. Broersma, F.V. Fomin, P.A. Golovach and G.J. Woeginger, Backbone colorings for networks, in: Proceedings of WG 2003, LNCS 2880 (2003) 131-142.
[4] H.J. Broersma, F.V. Fomin, P.A. Golovach and G.J. Woeginger, Backbone colorings for graphs: tree and path backbones, J. Graph Theory 55 (2007) 137-152, doi: 10.1002/jgt.20228.
[5] H.J. Broersma, J. Fujisawa, L. Marchal, D. Paulusma, A.N.M. Salman and K. Yoshimoto, λ-Backbone colorings along pairwise disjoint stars and matchings, preprint (2004). www.durham.ac.uk/daniel.paulusma/Publications/Papers/
Submitted/backbone.pdf
[6] H.J. Broersma, L. Marchal, D. Paulusma and A.N.M. Salman, Improved upper bounds for λ-backbone colorings along matchings and stars, in: Proceedings of the 33rd Conference on Current Trends in Theory and Practice of Computer Science SOFSEM 2007, LNCS 4362 (2007) 188-199.
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Received 17 December 2007
Accepted 23 October 2008