Geodesic
Grundy number of graphs
Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 1, pp. 5-18.

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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.
Keywords: Grundy coloring, vertex coloring, cartesian product, graph product 
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Effantin, Brice; Kheddouci, Hamamache. Grundy number of graphs. Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 1, pp. 5-18. http://geodesic.mathdoc.fr/item/DMGT_2007_27_1_a0/

[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.