Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, May 03, 2022, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Charlotte Knierim

Hat guessing numbers of strongly degenerate graphs

Assume n players are placed on the n vertices of a graph G. The following game was introduced by Winkler: An adversary puts a hat on each player, where each hat has a colour out of q available colours. The players can see the hat of each of their neighbours in G, but not their own hat. Using a prediscussed guessing strategy, the players then simultaneously guess the colour of their hat. The players win if at least one of them guesses correctly, else the adversary wins. The largest integer q such that there is a winning strategy for the players is denoted by HG(G), and this is called the hat guessing number of G.
Although this game has received a lot of attention in the recent years, not much is known about how the hat guessing number relates to other graph parameters. For instance, a natural open question is whether the hat guessing number can be bounded from above in terms of degeneracy. We show how the hat guessing number of a graph can be bounded from above in terms of a related notion, which we call strong degeneracy. We also give examples of graph classes with bounded strong degeneracy.
This is joint work with Anders Martinsson and Raphael Steiner

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