Bears with Hats and Independence Polynomials

TL;DR

The paper introduces the fractional hat chromatic number $\hat{\mu}$ for hat guessing games on graphs and establishes a deep link between winning strategies and the independence polynomial. By leveraging the monovariate polynomial $U_G$ and the Lovász Local Lemma, the authors show that for chordal graphs $\hat{\mu}(G)=1/r$ where $r$ is the smallest positive root of $U_G$, and they provide a polynomial-time algorithm to decide feasibility and to output winning strategies when feasible. They further develop a clique-join construction to build larger winning configurations from basic blocks (cliques), derive tight bounds $\hat{\mu}(G)=\Theta(\Delta)$ up to a $\log\Delta$ factor, and determine exact values for cliques, paths, and cycles, including the asymptotic limit $\lim_{n\to\infty}\hat{\mu}(P_n)=\lim_{n\to\infty}\hat{\mu}(C_n)=4$. The results fuse combinatorial game theory with probabilistic polynomial techniques, yielding both structural insights and practical algorithms for chordal graphs and key graph families.

Abstract

Consider the following hat guessing game. A bear sits on each vertex of a graph $G$, and a demon puts on each bear a hat colored by one of $h$ colors. Each bear sees only the hat colors of his neighbors. Based on this information only, each bear has to guess $g$ colors and he guesses correctly if his hat color is included in his guesses. The bears win if at least one bear guesses correctly for any hat arrangement. We introduce a new parameter - fractional hat chromatic number $\hatμ$, arising from the hat guessing game. The parameter $\hatμ$ is related to the hat chromatic number which has been studied before. We present a surprising connection between the hat guessing game and the independence polynomial of graphs. This connection allows us to compute the fractional hat chromatic number of chordal graphs in polynomial time, to bound fractional hat chromatic number by a function of maximum degree of $G$, and to compute the exact value of $\hatμ$ of cliques, paths, and cycles.

Figures (4)

• Figure 1: The clique join of graphs $G_1$ and $G_2$ with respect to $S$ and $v$.
• Figure 2: Applying Lemma \ref{['lem:CliqueJoin']} on winning hat guessing games $(C_4, 3, 1)$ (see Szczechla2017CycleGraphs) and $(K_3, 3, 1)$, we obtain a winning hat guessing game $(G, \mathbf{h}, 1)$ where $G$ is the result of identifying an edge in $C_4$ and $K_4$, and $\mathbf{h}$ is given in the picture.
• Figure 3: Application of Theorem \ref{['thm:Clique']} on a chordal graph $G$ with vector $\mathbf{r} \in \mathcal{R}(G)$. In each step, we highlight the clique $S$ and vertex $w$ that are used for Lemma \ref{['lem:CliqueJoin']} to inductively build a strategy for $G$ from strategies on cliques given by Theorem \ref{['thm:Clique']}. Note that the number of colors and guesses may differ from the depicted ratios by a multiplicative factor.
• Figure 4: A sketch of construction of the game $\mathcal{H}_{i+1}$. The formulas below vertices are the fractions $g_v/h_v$.

Theorems & Definitions (35)

• Proposition 1: folklore
• Theorem 1: Rational root theorem Larson07Calculus
• proof
• Lemma 1
• proof
• Theorem 2
• proof
• Corollary 1
• Lemma 2
• proof