On Intersection Representations and Clique Partitions of Graphs

A multifamily set representation of a finite simple graph G is a multifamily F of sets (not necessarily distinct) for which each set represents a vertex in G and two sets in F intersects if and only if the two corresponding vertices are adjacent. For a graph G, an edge clique covering (edge clique partition, respectively) Q is a set of cliques for which every edge is contained in at least (exactly, respectively) one member of Q. In 1966, P. Erdös, A. Goodman, and L. Pósa (The representation of a graph by set intersections, Canadian J. Math., 18, pp.106-112) pointed out that for a graph there is a one-to-one correspondence between multifamily set representations F and clique coverings Q for the edge set. Furthermore, for a graph one may similarly 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 1990, S. McGuinness and R. Rees (On the number of distinct minimal clique partitions and clique covers of a line graph, Discrete Math. 83 (1990) 49-62.) calculated the number of distinct Work supported partly by the National Science Council of Taiwan, Republic of China, under the Grant No. NSC 96-2115-M-029-001 The corresponding author with e-mail [email protected]