在双脊书本图上的恰当帽子猜测
Proper Hat-Guessing on Two-Spine Book Graphs
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中文总结 AI 辅助
研究在书本图\(B_{k,n}\)上的恰当帽子猜测游戏，给出覆盖性特征描述，证明\(\sup_{n\geq 1}\text{HGP}(B_{k,n}) = C_k\)等结论，确定\(C_2 = 11\)，解决部分有限情况并给出\(C_k\)的统一界限。
中文摘要 AI 辅助
在图上经典帽子猜测游戏的恰当变体中，对手用\(q\)种颜色的调色板为顶点恰当着色。每个顶点只能看到其邻居的颜色，并同时猜测自己的颜色；若至少有一个猜测正确，则玩家获胜。我们研究在书本图\(B_{k,n}=K_k\vee\overline{K_n}\)上的这个游戏，其中有\(k\)个相互相邻的脊顶点和\(n\)个独立的页面。我们给出了对每个固定脊大小都有效的覆盖性特征描述。设\(C_k\)是所有不可覆盖的恰当\(k\)元组的有限配置\(P\)中\(|P| + |\text{supp}(P)|\)的最小值。我们证明\(\sup_{n\geq 1}\text{HGP}(B_{k,n}) = C_k\)且对于所有足够大的\(n\)，\(\text{HGP}(B_{k,n}) = C_k\)。对于两个脊，覆盖性等同于伪森林性，我们精确地确定了相关的极值问题：\(C_2 = 11\)，有两种精确的极值阻碍类型。因此，对于每个\(n\)，\(\text{HGP}(B_{2,n})\leq 11\)，对于所有足够大的\(n\)等式成立；一个明确的概率估计给出了至多\(4\times 10^8\)的稳定阈值。我们还解决了前两个之前未解决的有限情况。一个具有仿射对称性的明确的七种颜色构造证明\(\text{HGP}(B_{2,3}) = 7\)。一个计数刚性论证表明对于所有\(n\geq 4\)，\(\text{HGP}(B_{2,n})\leq n + 3\)，这与单调性一起得出\(\text{HGP}(B_{2,4}) = 7\)。最后，一个一般的盒阻碍给出了\(C_k\)的明确统一界限。
英文摘要
In the proper variant of the classical hat-guessing game on a graph, an adversary properly colors the vertices from a palette of $q$ colors. Each vertex sees only its neighbors' colors and all vertices simultaneously guess their own color. The players win if at least one guess is correct. We study this game on the book graph $B_{k,n}=K_k\vee\overline{K_n}$, with $k$ mutually adjacent spine vertices and $n$ independent pages. We first give a coverability characterization valid for every fixed spine size. Write $\operatorname{HGP}(G)$ for the proper hat-guessing number of $G$. For a finite configuration $P$, let $\operatorname{supp}(P)$ denote the set of colors appearing in its tuples. Let $C_k$ be the minimum of $|P|+|\operatorname{supp}(P)|$ over all non-coverable finite configurations $P$ of proper $k$-tuples. We prove $\sup_{n\geq 1}\operatorname{HGP}(B_{k,n})=C_k$ and that $\operatorname{HGP}(B_{k,n})=C_k$ for all sufficiently large $n$. Thus, the asymptotic problem for every fixed $k$ reduces to a finite extremal invariant. In particular, coverability of two-spine configurations is equivalent to pseudoforestness, and we determine the associated extremal problem exactly: $C_2=11$, with precisely two types of extremal obstruction. Consequently, $\operatorname{HGP}(B_{2,n})\leq 11$ for every $n$, with equality for all sufficiently large $n$; we give an explicit probabilistic estimate with a stabilization threshold of at most $4\times 10^8$. We also resolve the first previously open finite cases. An explicit seven-color construction with affine symmetry proves $\operatorname{HGP}(B_{2,3})=7$. A counting-rigidity argument establishes the linear upper bound $\operatorname{HGP}(B_{2,n})\leq n+3$ for all $n\geq 4$, which together with monotonicity yields $\operatorname{HGP}(B_{2,4})=7$. Finally, a general box obstruction gives explicit uniform bounds on $C_k$.
