图的正常帽子猜测数的一个上界
An upper bound on the proper hat guessing number of graphs
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中文总结 AI 辅助
本文研究图的正常帽子猜测游戏，建立了依赖于顶点数和最大度的首个上界，并应用于随机图，证明其猜测数至多为 $cn$（$c\napprox1.366$），同时排除了高最大度图达到 $(2-o(1))n$ 的可能性。
中文摘要 AI 辅助
我们研究了由 Adriaensen 等人引入的图上的正常帽子猜测游戏（proper hat guessing game）。在该游戏中，玩家坐在图 $G$ 的顶点上，并从 $k$ 种颜色中分配帽子，使得分配结果构成一个正常着色（proper coloring）。每个顶点的可见性仅限于其邻域的帽子颜色。然后，他们必须同时输出关于自己帽子颜色的猜测。如果至少有一个猜测正确，则玩家获胜。与此问题相关的参数是正常帽子猜测数 $\noperatorname{HG}_{P}(G)$，即玩家能够保证获胜策略的最大颜色数 $m$。受 Shurman 等人工作的启发，我们建立了第一个同时依赖于顶点数 $n$ 和最大度 $\nDelta$ 的上界，适用于 $\nDelta \ngeq \frac{n}{e+1}$ 的情形。这一结果使我们证明二项随机图 $G_{n,1/2}$ 的正常帽子猜测数以 $cn$ 为上界，其中 $c \napprox 1.366$。最后，我们证明了对于某个固定的 $\ngamma \nin (0,1]$，最大度为 $(1-\ngamma)n$ 的图不可能有 $\noperatorname{HG}_{P}(G) = (2-o(1))n$。
英文摘要
We study the proper hat guessing game on graphs, introduced by Adriaensen et al. in Hat guessing with proper colorings. In this game, the players are seated on the vertices of a graph $G$ and assigned hats from a set of $k$ colors such that the resulting assignment forms a proper coloring. The visibility of each vertex is limited to the hat colors of their neighborhood. Then they must simultaneously output a guess about the color of their own hat. The players win if at least one guess is correct. A parameter related to this problem is the proper hat guessing number $\operatorname{HG}_{P}(G)$ that is the maximum number of colors $m$ such that the players can guarantee a winning strategy. Motivated by the work of Shurman et al. \cite{shurman2026upper}, we establish the first upper bound that depends both on the number of vertices $n$ and the maximum degree $Δ$ in the case where $Δ\geq \frac{n}{e+1}$. This result leads us to show that the proper hat guessing number of the binomial random graph $G_{n,1/2}$ is bounded above by $cn$, where $c \approx 1.366$. Finally, we prove that graphs of maximum degree $(1-γ)n$ for some fixed $γ\in (0,1]$ cannot have $\operatorname{HG}_{P}(G) = (2-o(1))n$.
发表机构
- Czech Technical University（捷克理工大学）
机构由 AI 辅助整理，请以论文原文为准。
