Cliques and constructors in “Hats” game. I

We analyze the following general variant of deterministic “Hats” game. Several sages wearing colored hats occupy the vertices of a graph, $k$-th sage can have hats of one of $h(k)$ colors. 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. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colors.
For complete graphs and for cycles we solve the problem of describing fuctions $h(k)$ for which the sages win. We develop “theory of construcors”, that is a collection of theorems demonstrating how one can construct new graphs for which the sages win. We define also new game “Check by rook” which is equivalent to Hats game on $4$-cycle and give complete analysis of this game.