Slavic Techniques for Hat Guessing Algorithms

TL;DR

This thesis explores the winnability of a deterministic hat guessing game on directed graphs, introducing new parameters, methods, and characterizations for cycles, trees, and general graphs, with several open problems posed.

Contribution

It provides new criteria and bounds for game winnability on various graph classes, generalizes existing methods, and introduces novel concepts like admissible paths and combinatorial prisms.

Findings

Cycle game winnability characterized by divisibility and sequence conditions.

Tree game winnability linked to subtree properties and exponential degree bounds.

Upper bounds on the parameters  and  for directed and undirected graphs.

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?