论文标题
帽子猜测书籍和风车
Hat Guessing on Books and Windmills
论文作者
论文摘要
戴上帽子的数字是由巴特勒，哈吉亚加伊，克莱恩伯格和莱顿定义的图形不变。我们确定了足够多页面的书籍图的帽子猜测数字，改善了以前已知的HE和LI下限，并与Gadouleau的上限完全匹配。我们证明，$ k_ {3,3} $的HAT猜测数字是$ 3 $，这使得这是第一个完整的两部分图$ k_ {n，n，n} $，其中已知HAT捕捞数量小于Gadouleau和Georgiou的$ N+1 $的上限。最后，我们确定了大多数参数选择的风车图数量。
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.
