# For which graphs the sages can guess correctly the color of at least one hat

## TL;DR

This paper characterizes the specific graphs where a group of sages can always correctly guess at least one hat color, given their neighbors' hats, using a predetermined strategy with three possible hat colors.

## Contribution

It provides a complete characterization of graphs that guarantee a winning guessing strategy for sages with three hat colors.

## Key findings

- Identifies all graphs where sages can guarantee at least one correct guess.
- Provides a complete solution to the guessing problem for three colors.
- Characterizes the structure of graphs enabling guaranteed success.

## Abstract

Several sages wearing colored hats occupy the vertices of a graph. Each sage tries to guess the color of his own hat merely on the basis of observing the hats of his neighbours without exchanging any information. Each hat can have one of three colors. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colors. We completely solve the problem of describing graphs for which the sages win.