We study the hat guessing game on graphs. In this game, a player is placed on each vertex $v$ of a graph $G$ and assigned a colored hat from $h(v)$ possible colors. Each player makes a deterministic guess on their hat color based on the colors assigned to the players on neighboring vertices, and the players win if at least one player correctly guesses his assigned color. If there exists a strategy that ensures at least one player guesses correctly for every possible assignment of colors, the game defined by $\langle G,h\rangle$ is called winning. The hat guessing number of $G$ is the largest integer $q$ so that if $h(v)=q$ for all $v\in G$ then $\langle G,h\rangle$ is winning. In this note, we determine whether $\langle G,h\rangle $ is winning for any $h$ whenever $G$ is a cycle, resolving a conjecture of Kokhas and Latyshev in the affirmative and extending it. We then use this result to determine the hat guessing number of every cactus graph, graphs in which every pair of cycles share at most one vertex.

翻译：我们研究图上的帽子猜测游戏。在该游戏中，每个顶点$v$上放置一名玩家，并分配一顶来自$h(v)$种可能颜色的帽子。每位玩家根据相邻顶点上玩家所分配的颜色对其自身帽子颜色进行确定性猜测；若至少有一名玩家正确猜出其被分配的颜色，则全体玩家获胜。若存在一种策略能确保对于所有可能的颜色分配至少有一名玩家正确猜测，则称由$\langle G,h\rangle$定义的游戏是获胜的。图$G$的帽子猜测数是最大的整数$q$，使得当对所有$v\in G$有$h(v)=q$时，$\langle G,h\rangle$是获胜的。在本文中，我们确定了当$G$为环时，对于任意$h$，$\langle G,h\rangle$是否获胜，从而肯定地回答并推广了Kokhas与Latyshev的猜想。随后我们利用这一结果确定了每个仙人掌图（即每对环至多共享一个顶点的图）的帽子猜测数。