Greedy Maximum-Clique Decompositions

A greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. We have recently shown that any greedy clique decomposition of a graph of order n has at most-~ cliques. A greedy max-clique decomposition is a particular kind of greedy clique decomposition where maximum cliques are removed, instead of just maximal ones. In this paper, we show that any greedy max-clique decomposition ~ of a n 2 graph of order n has ~ n(C) <_ ~-, where n(C) is the number of vertices in C.