### Clique coverings of graphs V: maximal-clique partitions

N.J. Pullman, H. Shank, W.D. Wallis
1982 Bulletin of the Australian Mathematical Society
A maximal-clique partition of a graph G is a way of covering G with maximal complete subgraphs, such that every edge belongs to exactly one of the subgraphs. If G has a maximal-clique partition, the maximal-clique partition number of G is the smallest cardinality of such partitions. In this paper the existence of maximal-clique partitions is discussed -for example, we explicitly describe all graphs with maximal degree at most four which have maximal-clique partitions -and discuss the
more » ... cuss the maximal-clique partition number and its relationship to other clique covering and partition numbers. The number of different maximal-clique partitions of a given graph is also discussed. Several open problems are presented. Received 9 Kovember 1981. We would like to thank Professor David A. Gregory for many helpful conversations. I is |C'| t |C| for all clique coverings C' of G , then C is called a minimum clique covering and |C| is called the clique covering number of G , denoted by cc(G) . Similarly cp(G) , the clique partition number of G , is the cardinality of a minimum clique partition. If G has no edges, we define cc(G) = cp(G) = 0 . Since every clique partition is also a clique covering we have cc(G) 5 cp(G) for all graphs G . Equality holds when (but not only when) G is triangle-free, in which case cc(G) = cp(G) = e(G) the number of edges of G . This subject has its origins in the problem of representing set intersections by graphs -see Erdos, Goodman and Posa [2], Lovasz [£], Harary [6] -and its matrix-theoretic form was studied by Ryser [15], [76]. Recently Orlin [JO], de Caen [J2], [73], Donald [?], [74] and Pullman [77],