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Efficient Graph Packing via Game Colouring

Published online by Cambridge University Press:  01 September 2009

H. A. KIERSTEAD  and
A. V. KOSTOCHKA
H. A. KIERSTEAD
Affiliation:
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA (e-mail: [email protected])
A. V. KOSTOCHKA
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA and Institute of Mathematics, Novosibirsk, Russia (e-mail: [email protected])
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Abstract

The game colouring number gcol(G) of a graph G is the least k such that, if two players take turns choosing the vertices of a graph, then either of them can ensure that every vertex has fewer than k neighbours chosen before it, regardless of what choices the other player makes. Clearly gcol(G) ≤ Δ(G)+1. Sauer and Spencer [20] proved that if two graphs G1 and G2 on n vertices satisfy 2Δ(G1)Δ(G2) < n then they pack, i.e., there is an embedding of G1 into the complement of G2. We improve this by showing that if (gcol(G1)−1)Δ(G2)+(gcol(G2)−1)Δ(G1) < n then G1 and G2 pack. To our knowledge this is the first application of colouring games to a non-game problem.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 5: New Directions in Algorithms, Combinatorics and Optimization , September 2009 , pp. 765 - 774
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Bodlaender, H. L. (1991) On the complexity of some coloring games. Internat. J. Found. Comput. Sci. 2 133–147.CrossRefGoogle Scholar
[2]Bollobás, B. and Eldridge, S. E. (1978) Packing of graphs and applications to computational complexity. J. Combin. Theory Ser. B 25 105–124.CrossRefGoogle Scholar
[3]Bollobás, B., Kostochka, A. V. and Nakprasit, K. (2005) On two conjectures on packing of graphs. Combin. Probab. Comput. 14 723–736.CrossRefGoogle Scholar
[4]Burr, S. A. and Erdős, P. (1975) On the magnitude of generalized Ramsey numbers. In Infinite and Finite Sets (A. Hajnal, R. Rado and V. T. Sós, eds), Colloq. Math. Soc. Janos Bolyai, Vol. 1, North-Holland.Google Scholar
[5]Catlin, P. A. (1974) Subgraphs of graphs I. Discrete Math. 10 225–233.Google Scholar
[6]Chen, G. and Schelp, R. H. (1993) Graphs with linearly bounded Ramsey numbers. J. Combin. Theory Ser. B 57 138–149.CrossRefGoogle Scholar
[7]Chvatál, V., Rödl, V., Szemerédi, E. and Trotter, W. T. (1983) The Ramsey number of a graph of bounded degree. J. Combin. Theory Ser. B 34 239–243.Google Scholar
[8]Dinski, T. and Zhu, X. (1999) A bound for the game chromatic number of graphs. Discrete Math. 196 109–115.Google Scholar
[9]Faigle, U., Kern, W., Kierstead, H. A. and Trotter, W. T. (1993) On the game chromatic number of some classes of graphs. Ars Combin. 35 143–150.Google Scholar
[10]Gardner, M. (1981) Mathematical games. Scientific American (April 1981), 23.Google Scholar
[11]Kierstead, H. A. (2000) A simple competitive graph coloring algorithm. J. Combin. Theory Ser. B 78 57–68.CrossRefGoogle Scholar
[12]Kierstead, H. A., Mohar, B., Špacapan, S., Yang, D. and Zhu, X. The two-coloring number and degenerate colorings of planar graphs. Submitted.Google Scholar
[13]Kierstead, H. A. and Trotter, W. T. (1994) Planar graph coloring with an uncooperative partner. J. Graph Theory 18 569–584.Google Scholar
[14]Kierstead, H. A. and Trotter, W. T. (2001) Competitive colorings of oriented graphs. Electron. J. Combin. 8 #12.Google Scholar
[15]Kierstead, H. A. and Tuza, Z. (2003) Marking games and the oriented game chromatic number of partial k-trees. Graphs Combin. 19 121–129.Google Scholar
[16]Kierstead, H. A. and Yang, D. (2003) Orderings on graphs and game coloring number. Order 20 255–264.CrossRefGoogle Scholar
[17]Kostochka, A. V. and Yu, G. (2007) An Ore-type analogue of the Sauer–Spencer theorem. Graphs Combin. 23 419–424.CrossRefGoogle Scholar
[18]Nešetřil, J. and Ossona de Mendez, P. (2008) Grad and classes with bounded expansion I: Decompositions. Europ. J. Combin. 29 760–776.CrossRefGoogle Scholar
[19]Nešetřil, J. and Sopena, E. (2001) On the oriented game chromatic number. Electron. J. Combin. 8 #14.CrossRefGoogle Scholar
[20]Sauer, N. and Spencer, J. (1978) Edge disjoint placement of graphs. J. Combin. Theory Ser. B 25 295–302.Google Scholar
[21]Zhu, X. (1999) Game coloring number of planar graphs. J. Combin. Theory Ser. B 75 245–258.Google Scholar
[22]Zhu, X. (2000) The game coloring number of pseudo partial k-trees. Discrete Math. 215 245–262.Google Scholar
[23]Zhu, X. (2008) Refined activation strategy for the marking game. J. Combin. Theory Ser. B 98 1–18.CrossRefGoogle Scholar
[24]Zhu, X. Colouring graphs with bounded generalized colouring number. Discrete Math., to appear.Google Scholar