Discussiones Mathematicae Graph Theory 27(1) (2007) 5-18
doi: 10.7151/dmgt.1339

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GRUNDY NUMBER OF GRAPHS

Brice Effantin and Hamamache Kheddouci

Laboratoire PRISMa, Université Claude Bernard Lyon 1
Bat. Nautibus (ex. 710), 843, Bd. du 11 novembre 1918
69622 Villeurbanne Cedex FRANCE
e-mail: {beffanti, hkheddou}@bat710.univ-lyon1.fr

Abstract

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.

2000 Mathematics Subject Classification: 05C15, 05C75.

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Received 19 September 2003
Revised 29 May 2006