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Grundy number and products of graphs

Marie Aste 1 Frédéric Havet 1 Claudia Linhares Sales 2 
1 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : The Grundy number of a graph G, denoted by Gamma(G), is the largest k such that G has a greedy k-colouring, that is a colouring with k colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we study the Grundy number of the lexicographic and cartesian products of two graphs in terms of the Grundy numbers of these graphs. Regarding the lexicographic product, we show that Gamma(G) x Gamma(H) <= Gamma(G[H]) <= 2(Gamma(G)-1)(Gamma(H) - 1) + Gamma(G). In addition, we show that if G is a tree or Gamma(G) = Delta(G) + 1, then Gamma(G[H]) = Gamma(G) x Gamma(H). We then deduce that for every fixed c >= 1, given a graph G, it is CoNP-Complete to decide if Gamma(G) <= c x chi (G) and it is CoNP-Complete to decide if Gamma(G) <= c x omega(G). Regarding the cartesian product, we show that there is no upper bound of Gamma(G square H) as a function of Gamma(G) and Gamma(H). Nevertheless, we prove that Gamma(G square H) <= Delta(G).2 Gamma((H)-1) + Gamma(H).
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https://hal.inria.fr/inria-00496038
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Submitted on : Tuesday, June 29, 2010 - 3:15:23 PM
Last modification on : Thursday, August 4, 2022 - 4:52:44 PM

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Marie Aste, Frédéric Havet, Claudia Linhares Sales. Grundy number and products of graphs. Discrete Mathematics, 2010, 310 (9), pp.1482-1490. ⟨10.1016/j.disc.2009.09.020⟩. ⟨inria-00496038⟩

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