Cliques and constructors in “Hats” game. II

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.
We demonstarte here winning strategies fo the sages on complete graphs, and analyze the Hats game on almost complete graphs. We prove also a collection of theorems demonstrating how one can construct new graphs for which the sages win.