Proper Hat-Guessing on Two-Spine Book Graphs

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 the colors of its neighbors and simultaneously guesses its 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 give a coverability characterization valid for every fixed spine size. 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 $\operatorname{HGP}(B_{k,n})=C_k$ for all sufficiently large $n$. Thus the asymptotic problem for fixed $k$ reduces to a finite extremal invariant. For two spines, coverability 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 $n$; an explicit probabilistic estimate gives a stabilization threshold of at most $4\times 10^8$. We also resolve the first two previously open finite cases. An explicit seven-color construction with affine symmetry proves $\operatorname{HGP}(B_{2,3})=7$. A counting-rigidity argument establishes $\operatorname{HGP}(B_{2,n})\leq n+3$ 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 $C_k$.