Discussiones Mathematicae Graph Theory 30(3) (2010) 499-533
doi: 10.7151/dmgt.1510

[BIBTex] [PDF] [PS]


Elżbieta Sidorowicz

Faculty of Mathematics, Computer Science and Econometrics
University of Zielona Góra
Z. Szafrana 4a, 65-516 Zielona Góra, Poland
e-mail: e.sidorowicz@wmie.uz.zgora.pl


For k ≥ 2 we define a class of graphs Hk = {G: every block of G has at most k vertices}. The class Hk contains among other graphs forests, Husimi trees, line graphs of forests, cactus graphs. We consider the colouring game and the generalized colouring game on graphs from Hk.

Keywords: colouring game, generalized colouring game.

2010 Mathematics Subject Classification: 05C15, 05C57.


[1] S.D. Andres, The game chromatic index of forests of maximum degree Δ ≥ 5, Discrete Applied Math. 154 (2006) 1317-1323, doi: 10.1016/j.dam.2005.05.031.
[2] H.L. Bodlaender, On the complexity of some colouring games, Internat. J. Found. Comput. Sci. 2 (1991) 133-148, doi: 10.1142/S0129054191000091.
[3] M. Borowiecki and P. Mihók, Hereditary properties of graphs, in: V.R. Kulli, ed(s), Advances in Graph Theory Vishwa International Publication Gulbarga, 1991) 41-68.
[4] M. Borowiecki and E. Sidorowicz, Generalized game colouring of graphs, Discrete Math. 307 (2007) 1225-1231, doi: 10.1016/j.disc.2005.11.060.
[5] L. Cai and X. Zhu, Game chromatic index of k-degenerate graphs, J. Graph Theory 36 (2001) 144-155, doi: 10.1002/1097-0118(200103)36:3<144::AID-JGT1002>3.0.CO;2-F.
[6] G. Chartrand and L. Leśniak, Graphs and Digraphs (Fourth Edition Chapman & Hall/CRC, 2005).
[7] Ch. Chou, W. Wang and X. Zhu, Relaxed game chromatic number of graphs, Discrete Math. 262 (2003) 89-98, doi: 10.1016/S0012-365X(02)00521-6.
[8] T. Dinski and X. Zhu, A bound for the game chromatic number of graphs, Discrete Math. 196 (1999) 109-115, doi: 10.1016/S0012-365X(98)00197-6.
[9] C. Dunn and H.A. Kierstead, The relaxed game chromatic number of outerplanar graphs, J. Graph Theory 46 (2004) 69-106, doi: 10.1002/jgt.10172.
[10] P.L. Erdös, U. Faigle, W. Hochstättler and W. Kern, Note on the game chromatic index of trees, Theoretical Computer Science 313 (2004) 371-376, doi: 10.1016/j.tcs.2002.10.002.
[11] U. Faigle, U. Kern, H.A. Kierstead and W.T. Trotter, On the game chromatic number of some classes of graphs, Ars Combin. 35 (1993) 143-150.
[12] D. Guan and X. Zhu, The game chromatic number of outerplanar graphs, J. Graph Theory 30 (1999) 67-70, doi: 10.1002/(SICI)1097-0118(199901)30:1<67::AID-JGT7>3.0.CO;2-M.
[13] W. He, J. Wu and X. Zhu, Relaxed game chromatic number of trees and outerplanar graphs, Discrete Math. 281 (2004) 209-219, doi: 10.1016/j.disc.2003.08.006.
[14] H.A. Kierstead, A simple competitive graph colouring algorithm, J. Combin. Theory (B) 78 (2000) 57-68, doi: 10.1006/jctb.1999.1927.
[15] H.A. Kierstead and Zs. Tuza, Marking games and the oriented game chromatic number of partial k-trees, Graphs Combin. 19 (2003) 121-129, doi: 10.1007/s00373-002-0489-5.
[16] E. Sidorowicz, The game chromatic number and the game colouring number of cactuses, Information Processing Letters 102 (2007) 147-151, doi: 10.1016/j.ipl.2006.12.003.
[17] J. Wu and X. Zhu, Lower bounds for the game colouring number of planar graphs and partial k-trees, Discrete Math. 308 (2008) 2637-2642, doi: 10.1016/j.disc.2007.05.023.
[18] J. Wu and X. Zhu, Relaxed game chromatic number of outerplanar graphs, Ars Combin. 81 (2006) 359-367.
[19] D. Yang and H.A. Kierstead, Asymmetric marking games on line graphs, Discrete Math. 308 (2008) 1751-1755, doi: 10.1016/j.disc.2007.03.082.
[20] X. Zhu, The Game Colouring Number of Planar Graphs, J. Combin. Theory (B) 75 (1999) 245-258, doi: 10.1006/jctb.1998.1878.
[21] X. Zhu, Game colouring number of pseudo partial k-trees, Discrete Math. 215 (2000) 245-262, doi: 10.1016/S0012-365X(99)00237-X.

Received 27 May 2010
Revised 28 June 2010
Accepted 28 June 2010