Discussiones Mathematicae Graph Theory 35 (2015) 157-169 doi: 10.7151/dmgt. 1786

OPTIMAL BACKBONE COLORING OF SPLIT GRAPHS WITH MATCHING BACKBONES

KRZYSZTOF TUROWSKI1

Gdansk University of Technology Department of Algorithms and System Modelling

e-mail: Krzysztof.Turowski@eti.pg.gda.pl.

Abstract

For a graph G with a given subgraph H, the backbone coloring is defined as the mapping c : V(G) ^ N+ such that |c(u) — c(v)| > 2 for each edge {u,v} e E(H) and |c(u) — c(v)| > 1 for each edge {u,v} e E(G). The backbone chromatic number BBC(G, H) is the smallest integer k such that there exists a backbone coloring with max„£V(G) c(v) = k.

In this paper, we present the algorithm for the backbone coloring of split graphs with matching backbone.

Keywords: backbone coloring, split graphs, matching. 2010 Mathematics Subject Classification: 05C15.

References

[1] P. Hammer and S. Foldes, Split graphs, Congr. Numer. XIX (1977) 311-315.

[2] J. Miskuf, R. Skrekovski and M. Tancer, Backbone colorings of graphs with bounded degree, Discrete Appl. Math. 158 (2010) 534-542. doi:10.1016/j.dam.2009.11.015

[3] H. 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

[4] H. Broersma, A general framework for coloring problems: old results, new results, and open problems, in: Combinatorial Geometry and Graph Theory: Indonesia-Japan Joint Conference, IJCCGGT 2003, Bandung, Indonesia, J. Akiyama, E.T. Baskoro, M. Kano (Ed(s)), (Springer, 2003) 65-79.

1This project has been partially supported by Narodowe Centrum Nauki under contract DEC-2011/02/A/ST6/00201.

[5] R. Janczewski, On an interrelation between travelling salesman problem and T-coloring of graphs, Proceedings of the Sixth International Conference: Advanced Computer Systems, ACS 1999, Szczecin, Poland (1999) 23-25.

Received 1 December 2011 Revised 18 November 2013 Accepted 1 May 2014

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