We investigate the problem of online learning a binary labeling of the vertices for a given graph. We design an algorithm, Majority, to solve the problem and show its optimality on clique graphs. For general graphs we derive a relevant mistake bound that relates the algorithm’s performance to the cut size (the number of edges between vertices with opposite labeling) and the maximum independent set in the graph. We next introduce a novel complexity measure of the true labeling - the frontier and relate the number of mistakes incurred by Majority to this measure. This allows us to show, in contrast to previous known approaches, that our algorithm works well even when the cut size is bigger than the number of vertices. A detailed comparison with previous results is given.