Hat Guessing Numbers of Degenerate Graphs

He, Xiaoyu; Li, Ray

doi:10.37236/9449

Cited by 10 publications

(30 citation statements)

References 11 publications

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“…The theorem above provides, of course, a counterexample to that as well. For larger Hadwiger numbers d the hat guessing number can in fact be at least doubly-exponential in d. This follows from the results in [10], as it is easy to show that the Hadwiger number of the graphs It is interesting to determine or estimate the hat guessing number of the random graph G = G(n, 1/2). In [4] it is shown that with high probability, that is, with probability tending to 1 as n tends to infinity,…”

Section: Introductionmentioning

confidence: 88%

“…This proves the claim and shows that in case all the vertices in the outer pairs guess incorrectly there are only 5 possible distinct colorings of the pair u, v. The vertices u, v see all hats of the vertices in the outer pairs and hence can compute this set of at most 5 distinct colorings of their hats. They can then take care of these colorings ensuring that at least one of them guesses correctly since as shown in [4,10] a clique of size 2 can handle any (known) set of 5 colorings. This shows that the hat guessing number of our graph is at least 12.…”

Section: Definition 13 ([1]mentioning

confidence: 99%

“…As shown in [4,10] there exists a set of 6 colorings, for example the set {0, 1} × {0, 1, 2}, such that the central vertices cannot guess correctly on all of them. Suppose the central pair gets one of these 6 colorings, and every outer pair gets a coloring in which both its members guess incorrectly for any of these 6 colorings.…”

Section: Definition 13 ([1]mentioning

confidence: 99%

“…The invariant HG(G) has been studied in several papers including [5,8,11,15,7,1,4,10,9]. In [4] the authors conjecture that the maximum possible hat guessing number of a planar graph is 4.…”

Section: Introductionmentioning

confidence: 99%

“…At that time this has been the planar graph with the largest hat guessing number known. Examples of planar graphs with hat guessing number 6 appear in [10] and in [9]. A more recent paper [12] contains a construction of a planar graph with hat guessing number 14.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 88%

Section: Definition 13 ([1]mentioning

confidence: 99%

Section: Definition 13 ([1]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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Bears with Hats and Independence Polynomials

Blažej

Dvořák

Opler

2021

Graph-Theoretic Concepts in Computer Science

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Consider the following hat guessing game. A bear sits on each vertex of a graph G, and a demon puts on each bear a hat colored by one of h colors. Each bear sees only the hat colors of his neighbors. Based on this information only, each bear has to guess g colors and he guesses correctly if his hat color is included in his guesses. The bears win if at least one bear guesses correctly for any hat arrangement.We introduce a new parameter—fractional hat chromatic number $$\hat{\mu }$$ μ ^ , arising from the hat guessing game. The parameter $$\hat{\mu }$$ μ ^ is related to the hat chromatic number which has been studied before. We present a surprising connection between the hat guessing game and the independence polynomial of graphs. This connection allows us to compute the fractional hat chromatic number of chordal graphs in polynomial time, to bound fractional hat chromatic number by a function of maximum degree of G, and to compute the exact value of $$\hat{\mu }$$ μ ^ of cliques, paths, and cycles.

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