  The hat-guessing number is a graph invariant defined by Butler, Hajiaghayi,
Kleinberg, and Leighton. We determine the hat-guessing number exactly for book
graphs with sufficiently many pages, improving previously known lower bounds of
He and Li and exactly matching an upper bound of Gadouleau. We prove that the
hat-guessing number of $K_{3,3}$ is $3$, making this the first complete
bipartite graph $K_{n,n}$ for which the hat-guessing number is known to be
smaller than the upper bound of $n+1$ of Gadouleau and Georgiou. Finally, we
determine the hat-guessing number of windmill graphs for most choices of
parameters.
