DMGT

Discussiones Mathematicae Graph Theory 24(2) (2004) 183-195
DOI: https://doi.org/10.7151/dmgt.1224

NEW LOWER BOUNDS ON THE WEIGHTED CHROMATIC NUMBER OF A GRAPH

Massimiliano Caramia

IAC - Istituto per le Applicazioni del Calcolo "M. Picone"
CNR - Viale del Policlinico, 137 - 00161 Roma, Italy
e-mail: caramia@iac.rm.cnr.it

Jirí Fiala

Institute of Theoretical Computer Science (ITI)
Charles University, Faculty of Mathematics and Physics
Malostranské nám. 2/25, 118 00, Prague, Czech Republic
e-mail: fiala@kam.mff.cuni.cz

Abstract

In this paper we present theoretical and algorithmic results for the computation of lower bounds on the chromatic number of a weighted graph. In particular, we study different ways of a possible improvement of the lower bound offered by a maximum weighted clique. Based on our findings we devise new algorithms and show their performance on random graphs.

Keywords: combinatorial analysis, computational analysis, optimization.

2000 Mathematics Subject Classification: 05C15, 05C85.

References

[1]	D. Brelaz, New methods to color the vertices of a graph, Communications of the ACM 22 (1979) 251-256, doi: 10.1145/359094.359101.
[2]	M. Caramia and P. Dell'Olmo, Iterative coloring extension of a maximum clique, Naval Research Logistics 48 (2001) 518-550, doi: 10.1002/nav.1033.
[3]	M. Caramia and P. Dell'Olmo, Solving the minimum weighted coloring problem, Networks 38 (2001) 88-101, doi: 10.1002/net.1028.
[4]	B. Descartes, Solution to advanced problem, No 4526. Amer. Math. Monthly 61 (1954) 532.
[5]	M.R. Garey and D.S. Johnson, Computers and Intractability (Freeman and Co.: San Francisco, 1979).
[6]	M. Kubale, Introduction to Computational Complexity and Algorithmic Graph Coloring (Wydawnictwo GTN, Gdańsk, 1998).
[7]	M. Kubale and B. Jackowski, A generalized implicit enumeration algorithm for graph coloring, Communications of the ACM 28 (1985) 412-418, doi: 10.1145/3341.3350.
[8]	A. Mehrotra and M.A. Trick, A column generation approach for graph coloring, INFORMS J. on Computing 8 (1996) 344-354, doi: 10.1287/ijoc.8.4.344.
[9]	T.J. Sager and S. Lin, A Pruning procedure for exact graph coloring, ORSA J. on Computing 3 (1993) 226-230, doi: 10.1287/ijoc.3.3.226.
[10]	E.C. Sewell, An Improved Algorithm for Exact Graph Coloring, in: D.S. Johnson and M.A. Trick, eds., DIMACS Series in Computer Mathematics and Theoretical Computer Science 26 (1996) 359-373.

Received 13 May 2002
Revised 27 October 2003