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On the Grundy number of a graph

Frédéric Havet 1 Leonardo Sampaio 1
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 $G$. Trivially $\Gamma(G)\leq \Delta(G)+1$. In this paper, we show that deciding if $\Gamma(G)\leq \Delta(G)$ is NP-complete. We then show that deciding if $\Gamma(G)\geq |V(G)|-k$ is fixed parameter tractable with respect to the parameter $k$.
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Frédéric Havet, Leonardo Sampaio. On the Grundy number of a graph. Fifth International Symposium on Parameterized and Exact Computation (IPEC 2010), Dec 2010, Chennai, India. pp.170--179, ⟨10.1007/978-3-642-17493-3⟩. ⟨inria-00532906⟩

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