On the hat problem on a graph - Publication - Bridge of Knowledge

Search

On the hat problem on a graph

Abstract

The topic of this paper is the hat problem in which each of n players is uniformly and independently fitted with a blue or red hat. Then everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The aim is to maximize the probability of winning. In this version every player can see everybody excluding himself. We consider such a problem on a graph, where vertices correspond to players, and a player can see each player to whom he isconnected by an edge. The solution of the hat problem on a graph is known for trees and for cycles on four or at least nine vertices. In this paper first we give an upper bound on the maximum chance of success for graphs with neighborhood-dominated vertices. Next we solve the problem on unicyclic graphs containing a cycle on at least nine vertices. We prove that the maximum chance of success is one by two. Then we consider the hat problem on a graph with a universal vertex. We prove that there always exists an optimal strategy such that in every case some vertex guesses its color. Moreover, we prove that there exists a graph with a universal vertex for which there exists an optimal strategy such that in some case no vertex guesses its color. We also give some Nordhaus-Gaddum type inequalities.

Cite as

Full text

download paper
downloaded 6 times
Publication version
Accepted or Published Version
License
Creative Commons: CC-BY open in new tab

Keywords

Details

Category:
Articles
Type:
artykuły w czasopismach recenzowanych i innych wydawnictwach ciągłych
Published in:
Opuscula Mathematica no. 32, pages 285 - 296,
ISSN: 1232-9274
Language:
English
Publication year:
2012
Bibliographic description:
Krzywkowski M.: On the hat problem on a graph// Opuscula Mathematica. -Vol. 32., iss. Iss. 2 (2012), s.285-296
Verified by:
Gdańsk University of Technology

seen 73 times

Recommended for you

Meta Tags