Cliques and Constructors in “Hats” Game. II

Kokhas, K. P.; Latyshev, Aleksei; Retinskiy, V. I.

doi:10.1007/s10958-021-05349-8

Cited by 7 publications

(10 citation statements)

References 6 publications

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“…We call constructors the theorems, which allow us to build new winning graphs by combining the graphs, for which their winning property is already proved. Here are several constructors from the papers [9,11,2]. E 2 ) be two graphs, S ⊆ G 1 be a clique, and v ∈ V 2 .…”

Section: Constructorsmentioning

confidence: 99%

“…Then A n are polynomials of k. Here we write down several polynomials A n , it is evident that they are closely related to Pascal triangle. Using (11) we can easily find the generating function of polynomials A n (k): It is easier to prove this fact than to find a reference. Remark that the number −4 is not really a root of A n (x) for any n.…”

Section: Denote It Bymentioning

confidence: 99%

“…In their previous works [9] and [11] (joint with V. Retinsky), the authors consider the version of the game with variable number of hats (i.e., the number of possible colors can be different from sage to sage). This version of game is not only of its own interest, but opens a more flexible approach to the classic Hats game analysis.…”

Section: Introductionmentioning

confidence: 99%

“…The rest of the paper is organized as follows. In section 2 we give necessary definitions and notations, recall several theorem-constructors from [9,11,2]. In section 3 in terms of independence polynomials we define maximal games and prove theorem that connects maximal games with constructors.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Hats game. On maximum degree and diameter

Latyshev,

Kokhas

2021

Preprint

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We analyze the following version of the deterministic Hats game. Several sages wearing colored hats occupy the vertices of a graph. Each sage can have a hat of one of k colors. Each sage tries to guess the color of his own hat merely on the basis of observing the hats of his neighbors without exchanging any information. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colors. The maximal number k for which the specific graph G is winning called hat guessing number (or HG(G)).We contradicted the famous hypothesis that HG(G) ≤ ∆ + 1 and proved that diameter of graph and HG(G) are independent.

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Section: Constructorsmentioning

confidence: 99%

Section: Denote It Bymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Hats game. On maximum degree and diameter

Latyshev,

Kokhas

2021

Preprint

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“…In the previous papers [6] and [7] (joint with V. Retinsky), the authors considered the version of the game with a variable number of hats (i.e., when the number of possible colors can differ from sage to sage). This version is not only of independent interest, but opens a more flexible approach to the analysis of the classical hat guessing game, because it has nontrivial techniques for building strategies.…”

Section: Introductionmentioning

confidence: 99%

The Hats Game. the Power of Constructors

Kokhas

Latyshev

2021

J Math Sci

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We analyze the following general version of the deterministic Hats game. Several sages wearing colored hats occupy the vertices of a graph. Each sage can have a hat of one of k colors. Each sage tries to guess the color of his own hat merely on the basis of observing the hats of his neighbors without exchanging any information. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colors.We present an example of a planar graph for which the sages win for k = 14. We also give an easy proof of the theorem about the Hats game on "windmill" graphs. Bibliography: 7 titles.

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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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Cliques and Constructors in “Hats” Game. II

Cited by 7 publications

References 6 publications

The Hats game. On maximum degree and diameter

The Hats game. On maximum degree and diameter

The Hats Game. the Power of Constructors

Bears with Hats and Independence Polynomials

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