Discussiones Mathematicae Graph Theory 29(2) (2009) 361-376
doi: 10.7151/dmgt.1452

[BIBTex] [PDF] [PS]


Krzysztof Giaro  and  Marek Kubale

Gdańsk University of Technology
Department of Algorithms and System Modeling
Narutowicza 11/12, 80-952 Gdańsk, Poland
e-mail: kubale@eti.pg.gda.pl


We consider a list cost coloring of vertices and edges in the model of vertex, edge, total and pseudototal coloring of graphs. We use a dynamic programming approach to derive polynomial-time algorithms for solving the above problems for trees. Then we generalize this approach to arbitrary graphs with bounded cyclomatic numbers and to their multicolorings.

Keywords: cost coloring, dynamic programming, list coloring, NP-completeness, polynomial-time algorithm.

2000 Mathematics Subject Classification: 05C15.


[1] K. Giaro and M. Kubale, Edge-chromatic sum of trees and bounded cyclicity graphs, Inf. Process. Lett. 75 (2000) 65-69, doi: 10.1016/S0020-0190(00)00072-7.
[2] K. Giaro, M. Kubale and P. Obszarski, A graph coloring approach to scheduling multiprocessor tasks on dedicated machines with availability constraints, Disc. Appl. Math., (to appear).
[3] S. Isobe, X. Zhou and T. Nishizeki, Cost total colorings of trees, IEICE Trans. Inf. and Syst. E-87 (2004) 337-342.
[4] J. Jansen, Approximation results for optimum cost chromatic partition problem, J. Alghoritms 34 (2000) 54-89, doi: 10.1006/jagm.1999.1022.
[5] M. Kao, T. Lam, W. Sung and H. Ting, All-cavity maximum matchings, Proc. ISAAC'97, LNCS 1350 (1997) 364-373.
[6] L. Kroon, A. Sen. H. Deng and A. Roy, The optimal cost chromatic partition problem for trees and interval graphs, Proc. WGTCCS'96, LNCS 1197 (1997) 279-292.
[7] D. Marx, The complexity of tree multicolorings, Proc. MFCS'02, LNCS 2420 (2002) 532-542.
[8] D. Marx, List edge muticoloring in graphs with few cycles, Inf. Proc. Lett. 89 (2004) 85-90, doi: 10.1016/j.ipl.2003.09.016.
[9] S. Micali and V. Vazirani, An O(mn1/2) algorithm for finding maximum matching in general graphs, Proc. 21st Ann. IEEE Symp. on Foundations of Computer Science (1980) 17-27.
[10] K. Mulmuley, U. Vazirani and V. Vazirani, Matching is as easy as matrix inversion, Combinatorica 7 (1987) 105-113, doi: 10.1007/BF02579206.
[11] T. Szkaliczki, Routing with minimum wire length in the dogleg-free Manhattan model is NP-complete, SIAM J. Computing 29 (1999) 274-287, doi: 10.1137/S0097539796303123.
[12] X. Zhou and T. Nishizeki, Algorithms for the cost edge-coloring of trees, LNCS 2108 (2001) 288-297.

Received 30 November 2007
Revised 26 February 2009
Accepted 26 February 2009