The game Grundy number of graphs

Havet, Frédéric; Zhu, Xuding

doi:10.1007/s10878-012-9513-8

Cited by 14 publications

(10 citation statements)

References 20 publications

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“…In order to tighten specific bounds for the game-chromatic number it may be necessary to design winning strategies for Alice that are not 'colourblind'. A further 'first-fit' variant of the graph colouring game is the Grundy colouring game introduced by Havet and Zhu (2013). The game-chromatic number has also been studied in the context of random graphs (Bohman et al, 2008).…”

Section: Motivationmentioning

confidence: 99%

Characterising and recognising game-perfect graphs

Andres,

Lock

2018

Preprint

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Consider a vertex colouring game played on a simple graph with k permissible colours. Two players, a maker and a breaker, take turns to colour an uncoloured vertex such that adjacent vertices receive different colours. The game ends once the graph is fully coloured, in which case the maker wins, or the graph can no longer be fully coloured, in which case the breaker wins. In the game gB, the breaker makes the first move. Our main focus is on the class of gB-perfect graphs: graphs such that for every induced subgraph H, the game gB played on H admits a winning strategy for the maker with only ω(H) colours, where ω(H) denotes the clique number of H. Complementing analogous results for other variations of the game, we characterise gB-perfect graphs in two ways, by forbidden induced subgraphs and by explicit structural descriptions. We also present a clique module decomposition, which may be of independent interest, that allows us to efficiently recognise gB-perfect graphs.

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Section: Motivationmentioning

confidence: 99%

Characterising and recognising game-perfect graphs

Andres,

Lock

2018

Preprint

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“…Finally, we address a question raised by Havet and Zhu [7,Problem 4] whether there is a function f such that all graphs G satisfy χ g (G) f (Γ g (G)). The following and the fact that Γ g (G) Γ (G) imply that the answer is no.…”

Section: Conjecturementioning

confidence: 99%

“…The Grundy game was introduced recently by Havet and Zhu [7], who showed that Γ g (F ) = 3 and Γ g (Q) 7. To our knowledge, the asymmetric Grundy games were not considered yet.…”

Section: Introductionmentioning

confidence: 99%

Asymmetric Coloring Games on Incomparability Graphs

Krawczyk

Walczak

2015

Electronic Notes in Discrete Mathematics

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Abstract. Consider the following game on a graph G: Alice and Bob take turns coloring the vertices of G properly from a fixed set of colors; Alice wins when the entire graph has been colored, while Bob wins when some uncolored vertices have been left. The game chromatic number of G is the minimum number of colors that allows Alice to win the game. The game Grundy number of G is defined similarly except that the players color the vertices according to the first-fit rule and they only decide on the order in which it is applied. The (a, b)-game chromatic and Grundy numbers are defined likewise except that Alice colors a vertices and Bob colors b vertices in each round. We study the behavior of these parameters for incomparability graphs of posets with bounded width. We conjecture a complete characterization of the pairs (a, b) for which the (a, b)-game chromatic and Grundy numbers are bounded in terms of the width of the poset; we prove that it gives a necessary condition and provide some evidence for its sufficiency. We also show that the game chromatic number is not bounded in terms of the Grundy number, which answers a question of Havet and Zhu.

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“…In 2013, Havet and Zhu [9] proposed the greedy coloring game and the game Grundy number Γ g (G). They proved that Γ g (G) ≤ 3 in forests and Γ g (G) ≤ 7 in partial 2-trees.…”

Section: Introductionmentioning

confidence: 99%

“…They also posed two questions. Problem 5 of [9]: χ g (G) can be bounded by a function of Γ g (G)? Problem 6 of [9]: Is it true that Γ g (G) ≤ χ g (G) for every graph G?…”

Section: Introductionmentioning

confidence: 99%

Hardness of some variants of the graph coloring game

Marcilon,

Martins,

Sampaio

2019

Preprint

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Very recently, a long-standing open question proposed by Bodlaender in 1991 was answered: the graph coloring game is PSPACEcomplete. In 2019, Andres and Lock proposed five variants of the graph coloring game and left open the question of PSPACE-hardness related to them.In this paper, we prove that these variants are PSPACE-complete for the graph coloring game and also for the greedy coloring game, even if the number of colors is the chromatic number. Finally, we also prove that a connected version of the graph coloring game, proposed by Charpentier et al. in 2019, is PSPACE-complete.

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The game Grundy number of graphs

Cited by 14 publications

References 20 publications

Characterising and recognising game-perfect graphs

Characterising and recognising game-perfect graphs

Asymmetric Coloring Games on Incomparability Graphs

Hardness of some variants of the graph coloring game

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