Abstract. Assume [Formula: see text] players are placed on [Formula: see text] vertices of a graph [Formula: see text]. The following game was introduced by Winkler: An adversary puts a hat on each player, where each hat has a color out of [Formula: see text] available colors. The players can see the hat of each of their neighbors in [Formula: see text] but cannot see their own hats. Using a predetermined guessing strategy, the players then simultaneously guess the color of their hats. The players win if at least one of them guesses correctly; otherwise, the adversary wins. The largest integer [Formula: see text] such that there is a winning strategy for the players is denoted by [Formula: see text], and this is called the hat guessing number of [Formula: see text]. Although this game has received much attention in 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. In this paper, we prove that the hat guessing number of a graph can be bounded from above in terms of a related notion, which we call strong degeneracy. We further give an exact characterization of graphs with bounded strong degeneracy. As a consequence, we significantly improve the best known upper bound on the hat guessing number of outerplanar graphs from [Formula: see text] to 40 and further derive upper bounds on the hat guessing number for any class of [Formula: see text]-free graphs with bounded expansion, such as the class of [Formula: see text]-free planar graphs; more generally, for [Formula: see text]-free graphs with bounded Hadwiger number or without a [Formula: see text]-subdivision; and for Erdős–Rényi random graphs with constant average degree.

In the Maximum Label Propagation Algorithm (Max-LPA), each vertex draws a distinct random label. In each subsequent round, each vertex updates its label to the label that is most frequent among its neighbours (including its own label), breaking ties towards the larger label. It is known that this algorithm can detect communities in random graphs with planted communities if the graphs are very dense, by converging to a different consensus for each community. In [Kothapalli et al., 2013] it was also conjectured that the same result still holds for sparse graphs if the degrees are at least C log n. We disprove this conjecture by showing that even for degrees n^epsilon, for some epsilon>0, the algorithm converges without reaching consensus. In fact, we show that the algorithm does not even reach almost consensus, but converges prematurely resulting in orders of magnitude more communities.