The Hat Guessing Number of Graphs

Noga Alon, Omri Ben-Eliezer, Chong Shangguan, Itzhak Tamo

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Scopus citations

Abstract

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 Kn,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 Kn,...,n satisfies {\text{HG}}({K-{n, \ldots ,\;n}}) = \Omega \left( {{n^{\frac{{r - 1}}{r} - o(1)}}} \right). Interestingly, our guessing strategy is based on a combinatorial construction related to the zero-error list-decoding code for the q/(q - 1) channel.Additionally, we consider related problems like the relation between the hat guessing number and other graph parameG}}({K-{n, \ldots ,\;n}}) = \Omega \left( {{n^{\frac{{r - 1}}{r} - o(1)}}} \right). Interestingly, our guessing strategy is based on a combinatorial construction related to the zero-error list-decoding code for the q/(q - 1) channel.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. Among other results, it is shown that under certain conditions, the linear hat guessing number of Kn,n is at most 3, exhibiting a huge gap from the \Omega \left( {{n^{\frac{1}{2} - o(1)}}} \right) (nonlinear) hat guessing number of this graph.

Original languageEnglish (US)
Title of host publication2019 IEEE International Symposium on Information Theory, ISIT 2019 - Proceedings
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages490-494
Number of pages5
ISBN (Electronic)9781538692912
DOIs
StatePublished - Jul 2019
Event2019 IEEE International Symposium on Information Theory, ISIT 2019 - Paris, France
Duration: Jul 7 2019Jul 12 2019

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
Volume2019-July
ISSN (Print)2157-8095

Conference

Conference2019 IEEE International Symposium on Information Theory, ISIT 2019
CountryFrance
CityParis
Period7/7/197/12/19

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Information Systems
  • Modeling and Simulation
  • Applied Mathematics

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    Alon, N., Ben-Eliezer, O., Shangguan, C., & Tamo, I. (2019). The Hat Guessing Number of Graphs. In 2019 IEEE International Symposium on Information Theory, ISIT 2019 - Proceedings (pp. 490-494). [8849500] (IEEE International Symposium on Information Theory - Proceedings; Vol. 2019-July). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ISIT.2019.8849500