On the hat problem on a graph

Krzywkowski, Marcin

doi:10.7494/opmath.2012.32.2.285

Cited by 10 publications

(14 citation statements)

References 14 publications

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“…We say that a graph G is a subgraph of G if it can be obtained from G by using only the operations of removing edges and removing vertices. The following proposition, which was already used in previous parts of our paper and in earlier work [7], is presented for completeness.…”

Section: Graph Transformationsmentioning

confidence: 96%

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On Optimal Strategies for a Hat Game on Graphs

Feige¹

2010

SIAM J. Discrete Math.

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The following problem was introduced by Marcin Krzywkowski as a generalization of a problem of Todd Ebert. After initially coordinating a strategy, n players each occupy a different vertex of a graph. Either blue or red hats are placed randomly and independently on their heads. Each player sees the colors of the hats of players in neighboring vertices and no other hats (and hence, in particular, the player does not see the color of his own hat). Simultaneously, each player either tries to guess the color of his own hat or passes. The players win if at least one player guesses correctly and no player guesses wrong. The value of the game is the winning probability of the strategy that maximizes this probability. Previously, the value of such games was derived for certain families of graphs, including complete graphs of carefully chosen sizes, trees, and the 4-cycle. In this manuscript we conjecture that on every graph there is an optimal strategy in which all players who do not belong to the maximum clique always pass. We provide several results that support this conjecture, and determine among other things the value of the hat game for any bipartite graph and any planar graph that contains a triangle.

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Section: Graph Transformationsmentioning

confidence: 96%

“…If the graph is a complete graph, this is exactly Ebert's original problem. In [7] it is shown that if the graph is a tree, the value of the corresponding game is 1/2. In [8] the same result is shown when the graph is C 4 (a cycle on four vertices).…”

Section: Introductionmentioning

confidence: 99%

On Optimal Strategies for a Hat Game on Graphs

Feige¹

2010

SIAM J. Discrete Math.

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“…This version has been investigated further by Gadouleau and Georgiou [18], Szczechla [31] and Gadouleau [17]. The reader is referred to the theses of Farnik [14] and Krzywkowski [24] for extensive reviews on different hat guessing games.…”

Section: Related Workmentioning

confidence: 99%

The Hat Guessing Number of Graphs

Alon

Ben‐Eliezer

Shangguan

et al. 2019

2019 IEEE International Symposium on Information Theory (ISIT)

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Consider the following hat guessing game: n players are placed on n vertices of a graph, each wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. Given a graph G, its hat guessing number HG(G) is the largest integer q such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of q possible colors.In 2008, Butler et al. asked whether the hat guessing number of the complete bipartite graph K n,n is at least some fixed positive (fractional) power of n. We answer this question affirmatively, showing that for sufficiently large n, the complete r-partite graph K n,...,n satisfies HG(K n,...,n ) = Ω(n r−1 r −o(1) ). Our guessing strategy is based on a probabilistic construction and other combinatorial ideas, and can be extended to show that HG( C n,...,n ) = Ω(n 1 r −o(1) ), where C n,...,n is the blow-up of a directed r-cycle, and where for directed graphs each player sees only the hat colors of his outneighbors.Additionally, we consider related problems like the relation between the hat guessing number and other graph parameters, and the linear hat guessing number, where the players are only allowed to use affine linear guessing strategies. Several nonexistence results are obtained by using well-known combinatorial tools, including the Lovász Local Lemma and the Combinatorial Nullstellensatz. Among other results, it is shown that under certain conditions, the linear hat guessing number of K n,n is at most 3, exhibiting a huge gap from the Ω(n 1 2 −o(1) ) (nonlinear) hat guessing number of this graph.

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“…The first graph is trivial. For the directed cycle on vertices 1, 2, 3 and arcs (1, 2), (2, 3), (3,1), the function f is…”

Section: The Graph D On Six Vertices Inmentioning

confidence: 99%

New Constructions and Bounds for Winkler's Hat Game

Gadouleau¹,

Georgiou²

2015

SIAM J. Discrete Math.

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Hat problems have recently become a popular topic in combinatorics and discrete mathematics. These have been shown to be strongly related to coding theory, network coding, and auctions. We consider the following version of the hat game, introduced by Winkler and studied by Butler et al. A team is composed of several players; each player is assigned a hat of a given colour; they do not see their own colour, but can see some other hats, according to a directed graph. The team wins if they have a strategy such that, for any possible assignment of colours to their hats, at least one player guesses their own hat colour correctly. In this paper, we discover some new classes of graphs which allow a winning strategy, thus answering some of the open questions in Butler et al. We also derive upper bounds on the maximal number of possible hat colours that allow for a winning strategy for a given graph.

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On the hat problem on a graph

Cited by 10 publications

References 14 publications

On Optimal Strategies for a Hat Game on Graphs

On Optimal Strategies for a Hat Game on Graphs

The Hat Guessing Number of Graphs

New Constructions and Bounds for Winkler's Hat Game

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