On Greedy Clique Decompositions and Set Representations of Graphs

In 1994 S. McGuinness showed that any greedy clique decomposition of an n-vertex graph has at most ⌊n2/4⌋ cliques (The greedy clique decomposition of a graph, J. Graph Theory 18 (1994) 427-430), where a clique decomposition means a clique partition of the edge set and a greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. This result solved a conjecture by P. Winkler. A multifamily set representation of a simple graph G is a family of sets, not necessarily distinct, each member of which represents a vertex in G, and the intersection of two sets is non-empty if and only if two corresponding vertices in G are adjacent. It is well known that for a graph G, there is a one-to-one correspondence between multifamily set representations and clique coverings of the edge set. Further for a graph one may have a one-to-one correspondence between particular multifamily set representations with intersection size at most one and clique partitions of the edge set. In this paper, we study for an n-vertex graph the variant of the set representations using a family of distinct sets, including the greedy way to get the corresponding clique partition of the edge set of the graph. Similarly, in this case, we obtain a result that any greedy clique decomposition of an n-vertex graph has at most ⌊n2/4⌋ cliques. Supported partially by the National Science Council under Grants NSC 96-2115-M029-001 and NSC 96-2115-M-029-007 E-mail address: [email protected]