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2007 | 27 | 1 | 5-18
Tytuł artykułu

Grundy number of graphs

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The Grundy number of a graph G is the maximum number k of colors used to color the vertices of G such that the coloring is proper and every vertex x colored with color i, 1 ≤ i ≤ k, is adjacent to (i-1) vertices colored with each color j, 1 ≤ j ≤ i -1. In this paper we give bounds for the Grundy number of some graphs and cartesian products of graphs. In particular, we determine an exact value of this parameter for n-dimensional meshes and some n-dimensional toroidal meshes. Finally, we present an algorithm to generate all graphs for a given Grundy number.
Wydawca
Rocznik
Tom
27
Numer
1
Strony
5-18
Opis fizyczny
Daty
wydano
2007
otrzymano
2003-09-19
poprawiono
2006-05-29
Twórcy
  • Laboratoire PRISMa, Université Claude Bernard Lyon 1, Bat. Nautibus (ex. 710), 843, Bd. du 11 novembre 1918, 69622 Villeurbanne Cedex, France
  • Laboratoire PRISMa, Université Claude Bernard Lyon 1, Bat. Nautibus (ex. 710), 843, Bd. du 11 novembre 1918, 69622 Villeurbanne Cedex, France
Bibliografia
  • [1] C.Y. Chang, W.E. Clark and E.O. Hare, Domination Numbers of Complete Grid Graphs, I, Ars Combinatoria 36 (1994) 97-111.
  • [2] C.A. Christen and S.M. Selkow, Some perfect coloring properties of graphs, J. Combin. Theory B27 (1979) 49-59.
  • [3] N. Cizek and S. Klavžar, On the chromatic number of the lexicographic product and the cartesian sum of graphs, Discrete Math. 134 (1994) 17-24, doi: 10.1016/0012-365X(93)E0056-A.
  • [4] J.E. Dunbar, S.M. Hedetniemi, S.T. Hedetniemi, D.P. Jacobs, J. Knisely, R.C. Laskar and D.F. Rall, Fall Colorings of Graphs, J. Combin. Math. and Combin. Computing 33 (2000) 257-273.
  • [5] C. Germain and H. Kheddouci, Grundy coloring for power caterpillars, Proceedings of International Optimization Conference INOC 2003, 243-247, October 2003 Evry/Paris France.
  • [6] C. Germain and H. Kheddouci, Grundy coloring of powers of graphs, to appear in Discrete Math. 2006.
  • [7] S. Gravier, Total domination number of grid graphs, Discrete Appl. Math. 121 (2002) 119-128, doi: 10.1016/S0166-218X(01)00297-9.
  • [8] S.M. Hedetniemi, S.T. Hedetniemi and T. Beyer, A linear algorithm for the Grundy (coloring) number of a tree, Congr. Numer. 36 (1982) 351-363.
  • [9] J.D. Horton and W.D. Wallis, Factoring the cartesian product of a cubic graph and a triangle, Discrete Math. 259 (2002) 137-146, doi: 10.1016/S0012-365X(02)00376-X.
  • [10] M. Kouider and M. Mahéo, Some bound for the b-chromatic number of a graph, Discrete Math. 256 (2002) 267-277, doi: 10.1016/S0012-365X(01)00469-1.
  • [11] A. McRae, NP-completeness Proof for Grundy Coloring, unpublished manuscript 1994. Personal communication.
  • [12] C. Parks and J. Rhyne, Grundy Coloring for Chessboard Graphs, Seventh North Carolina Mini-Conference on Graph Theory, Combinatorics, and Computing (2002).
  • [13] F. Ruskey and J. Sawada, Bent Hamilton Cycles in d-Dimensional Grid Graphs, The Electronic Journal of Combinatorics 10 (2003) #R1.
  • [14] J.A. Telle and A. Proskurowski, Algorithms for vertex partitioning problems on partial k-trees, SIAM J. Discrete Math. 10 (4) (1997) 529-550, doi: 10.1137/S0895480194275825.
  • [15] X. Zhu, Star chromatic numbers and products of graphs, J. Graph Theory 16 (1992) 557-569, doi: 10.1002/jgt.3190160604.
  • [16] X. Zhu, The fractional chromatic number of the direct product of graphs, Glasgow Mathematical Journal 44 (2002) 103-115, doi: 10.1017/S0017089502010066.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1339
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