Hat-guessing on graphs

Hat-guessing games are combinatorial puzzles in which people try to guess the colors of their own hats. In the variant we study, each person is assigned 1 of q possible hat colors and is placed at the vertex on a graph G. Players can only see the hat colors of the people at adjacent vertices, but not their own. The players simultaneously guess the colors of their own hats. What is HGpGq, the largest q for which the players on G can devise a strategy that guarantees at least one person guesses correctly? We find the exact values of HGpGq for the following graphs. HG pK3,3q “ 3. For windmill graphs, HGpWdpk, nqq “ 2k ́ 2 for n ě log2p2k ́ 2q. For book graphs, HG pBd,nq “ 1 ` řd i“1 i i for sufficiently large n. Finally, we determine the hat-guessing number for all 5-vertex undirected graphs.