Discussiones Mathematicae Graph Theory 26(3) (2006) 369-376
doi: 10.7151/dmgt.1329

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THE COST CHROMATIC NUMBER AND HYPERGRAPH PARAMETERS

Gábor Bacsó  and  Zsolt Tuza

Computer and Automation Institute
Hungarian Academy of Sciences
H-1111 Budapest, Kende u. 13-17, Hungary

Abstract

In a graph, by definition, the weight of a (proper) coloring with positive integers is the sum of the colors. The chromatic sum is the minimum weight, taken over all the proper colorings. The minimum number of colors in a coloring of minimum weight is the cost chromatic number or strength of the graph. We derive general upper bounds for the strength, in terms of a new parameter of representations by edge intersections of hypergraphs.

Keywords: graph coloring, cost chromatic number, intersection number of a hypergraph.

2000 Mathematics Subject Classification: Primary: 05C15, 05C62; Secondary: 05C35, 05C65.

References

[1] Y. Alavi, P. Erdős, P.J. Malde, A.J. Schwenk and C. Thomassen, Tight bounds on the chromatic sum of a connected graph, J. Graph Theory 13 (1989) 353-357, doi: 10.1002/jgt.3190130310.
[2] U. Feigle, L. Lovász and P. Tetali, Approximating sum set cover, Algorithmica 40 (2004) 219-234, doi: 10.1007/s00453-004-1110-5.
[3] M. Gionfriddo, F. Harary and Zs. Tuza, The color cost of a caterpillar, Discrete Math. 174 (1997) 125-130, doi: 10.1016/S0012-365X(96)00325-1.
[4] E. Kubicka, Constraints on the chromatic sequence for trees and graphs, Congr. Numer. 76 (1990) 219-230.
[5] E. Kubicka and A.J. Schwenk, An introduction to chromatic sums, in: Proc. ACM Computer Science Conference, Louisville (Kentucky) 1989, 39-45.
[6] J. Mitchem and P. Morriss, On the cost chromatic number of graphs, Discrete Math. 171 (1997) 201-211, doi: 10.1016/S0012-365X(96)00005-2.
[7] J. Mitchem, P. Morriss and E. Schmeichel, On the cost chromatic number of outerplanar, planar and line graphs, Discuss. Math. Graph Theory 17 (1997) 229-241, doi: 10.7151/dmgt.1050.
[8] M. Molloy and B. Reed, Graph Colouring and the Probabilistic Method (Springer, 2002).
[9] T. Szkaliczki, Routing with minimum wire length in the Dogleg-free Manhattan Model, SIAM Journal on Computing 29 (1999) 274-287, doi: 10.1137/S0097539796303123.
[10] Zs. Tuza, Contractions and minimal k-colorability, Graphs and Combinatorics 6 (1990) 51-59, doi: 10.1007/BF01787480.
[11] Zs. Tuza, Problems and results on graph and hypergraph colorings, Le Matematiche 45 (1990) 219-238.

Received 1 December 2005
Revised 19 June 2006