Slavic Techniques for Hat Guessing Algorithms

Abstract

2023 undergraduate thesis on a deterministic "hat game." For a digraph $D$, each player stands on a vertex $v$, is assigned a hat from $h(v)$ possible colors, and makes $g(v)$ guesses of her hat's color based on her out-neighbors' hats. If there exists a collective strategy that guarantees a correct guess for any hat assignment, the game is winnable. Which games $(D,g,h)$ are winnable? Two much-studied parameters: $\mu(D)$ is the maximum integer $k$ such that $(D,1,k)$ is winnable, and $\hat{\mu}(D)$ is the supremum of $h/g$ for integer $h, g$ such that $(D,g,h)$ is winnable. Chapter 0 is a casual, riddle-based introduction. Chapter 1 taxonomizes the games, surveys all previous work, and summarizes the piece. Chapter 2 proves lemmata and easy cases. Chapter 3 uses "hats as hints" and "admissible paths" for games on cycles. Chapter 4 generalizes several "constructors" and applies them to tree games. Chapter 5 uses "combinatorial prisms" for a new angle on the well-studied $K_{n,m}$ games. In chapter 6, we apply "dependency digraphs" to the continuous limit of this game. Chapter 7 collects open problems and minor results. We show: $(C_{k\geq 4},1,h)$ is winnable if and only if: $h=3$ and $k$ is divisible by $3$ or equal to $4$, $h\leq 4$ and the $h(v)$ sequence $(3,2,3)$ or $(2,3,3)$ appears in the cycle, or the $h(v)$ sequence $(2,...,2)$ appears with no intervening value $>4$. $(T, 1, h)$ is winnable for tree $T$ iff $T$ has a subtree $T'$ with $h(v)\leq 2^{deg_{T'}(v)}$ for all $v\in V(T')$. For a digraph $D$, $\hat{\mu}(D)\leq e(\Delta^-+1)$. For a graph $G$, $\hat{\mu}(G)\leq (\Delta-1)^{1-\Delta} \Delta^{\Delta}<e\Delta$. $(\overrightarrow{C}_k, g, h)$ is unwinnable if $g(v_i)/h(v_i) + g(v_{i+1})/h(v_{i+1}) < 1$ for some $i$. And much else. Important open questions: what other graph parameters or properties bound $\mu$? What complexity classes are at play?