The hat guessing number of graphs

Alon, Noga; Ben‐Eliezer, Omri; Shangguan, Chong; Tamo, Itzhak

doi:10.1016/j.jctb.2020.01.003

Cited by 19 publications

(22 citation statements)

References 16 publications

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“…Taking G = B d,n , we see that the spine of the book has exactly d vertices and is a vertex cover for G, so HG(B d,n ) = 1 + h(N d ) satisfies (2). For the lower bound, as h(N) = 1, it suffices to prove for d 2 that (vacuously, no value is 0-abundant), and that x ∈ N d is type-i if i is the smallest index such that x i is not i-abundant, and say x is type-0 if it is not type-i for any i ∈ [d].…”

Section: Booksmentioning

confidence: 99%

“…[19,20]). In this note, we study a particular hat puzzle on graphs introduced by Butler, Hajiaghayi, Kleinberg, and Leighton [5], which has attracted some recent interest in discrete mathematics [2,3,8,9,10,12,18].…”

Section: Introductionmentioning

confidence: 99%

“…This was later disproved by Bosek,Dudek,Farnik,Grytczuk,and Mazur [3], who showed that for n large enough, HG(B d,n ) 2 d for the so-called book graph B d,n , which consists of a d-clique complete to n isolated vertices, and which is d-degenerate. 1 Alon, Ben-Eliezer, Shangguan, and Támo [2] considered another variant of this question, showing that if q is a prime power and the players' strategies are constrained to be linear (identifying the q hat colors with elements of F q ), then the players cannot win unless q d + 1.…”

Section: Introductionmentioning

confidence: 99%

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Hat Guessing Numbers of Degenerate Graphs

2020

Electron. J. Combin.

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Recently, Farnik asked whether the hat guessing number $\mathrm{HG}(G)$ of a graph $G$ could be bounded as a function of its degeneracy $d$, and Bosek, Dudek, Farnik, Grytczuk and Mazur showed that $\mathrm{HG}(G)\ge 2^d$ is possible. We show that for all $d\ge 1$ there exists a $d$-degenerate graph $G$ for which $\mathrm{HG}(G) \ge 2^{2^{d-1}}$. We also give a new general method for obtaining upper bounds on $\mathrm{HG}(G)$. The question of whether $\mathrm{HG}(G)$ is bounded as a function of $d$ remains open.

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Section: Booksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hat Guessing Numbers of Degenerate Graphs

2020

Electron. J. Combin.

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“…The hat-guessing numbers of graphs other than the complete graph have proven surprisingly difficult to compute. The value of HG(G) has been determined for trees [7], cycles [16], extremely unbalanced complete bipartite graphs [2], and certain tree-like degenerate graphs [11], but outside of these very specific families little is known. In this paper we add to this list of solved graphs almost all books and windmills, as well as the graph K 3,3 , which is in some sense the first "interesting" complete bipartite graph for this problem.…”

Section: Introductionmentioning

confidence: 99%

“…In the paper defining the hat-guessing game [7], it was proved that for large n, HG (K n,n ) = Ω(log log n). Later, Gadouleau and Georgiou [9] proved that Ω(log n) HG (K n,n ) n + 1, and most recently, Alon, Ben-Eliezer, Shangguan, and Tamo [2] improved the lower bound to HG (K n,n ) = Ω(n…”

Section: Introductionmentioning

confidence: 99%

Hat Guessing on Books and Windmills

Ido

Benjamin

2022

Electron. J. Combin.

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The hat-guessing number is a graph invariant defined by Butler, Hajiaghayi, Kleinberg, and Leighton. We determine the hat-guessing number exactly for book graphs with sufficiently many pages, improving previously known lower bounds of He and Li and exactly matching an upper bound of Gadouleau. We prove that the hat-guessing number of $K_{3,3}$ is $3$, making this the first complete bipartite graph $K_{n,n}$ for which the hat-guessing number is known to be smaller than the upper bound of $n+1$ of Gadouleau and Georgiou. Finally, we determine the hat-guessing number of windmill graphs for most choices of parameters.

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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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The hat guessing number of graphs

Cited by 19 publications

References 16 publications

Hat Guessing Numbers of Degenerate Graphs

Hat Guessing Numbers of Degenerate Graphs

Hat Guessing on Books and Windmills

Bears with Hats and Independence Polynomials

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