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

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.