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

Several sages wearing coloured hats are situated at the vertices of a graph. Each sage tries to guess his own hat colour merely on the basis of observing the hats worn by their neighbours without exchanging the information. Each hat can have one of three colours. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colours. We completely solve the question for which graphs the sages win.