Wyszukiwarka
Abstrakt
A team of n players plays the following game. After a strategy session, each player is randomly fitted with a blue or red hat. Then, without further communication, everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. Visibility is defined by a directed graph; that is, vertices correspond to players, and a player can see each player to whom he is connected by an arc. 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 team aims to maximize the probability of a win, and this maximum is called the hat number of the graph.Previous works focused on the hat problem on complete graphs and on undirected graphs. Some cases were solved, e.g., complete graphs of certain orders, trees, cycles, and bipartite graphs. These led Uriel Feige to conjecture that the hat number of any graph is equal to the hat number of its maximum clique.We show that the conjecture does not hold for directed graphs. Moreover, for every value of the maximum clique size, we provide a tight characterization of the range of possible values of the hat number. We construct families of directed graphs with a fixed clique number the hat number of which is asymptotically optimal. We also determine the hat number of tournaments to be one half.
Cytuj jako
Pełna treść
- Wersja publikacji
- Accepted albo Published Version
- Licencja
-
otwiera się w nowej karcie
Słowa kluczowe
Informacje szczegółowe
- Kategoria:
- Publikacja w czasopiśmie
- Typ:
- artykuł w czasopiśmie wyróżnionym w JCR
- Opublikowano w:
-
ELECTRONIC JOURNAL OF COMBINATORICS
nr 19,
ISSN: 1077-8926 - Język:
- angielski
- Rok wydania:
- 2012
- Opis bibliograficzny:
- Hod R., Krzywkowski M.: A construction for the hat problem on a directed graph// ELECTRONIC JOURNAL OF COMBINATORICS. -Vol. 19, (2012),
- Weryfikacja:
- Politechnika Gdańska
wyświetlono 112 razy