On the number of complete subgraphs and circuits in a graph

Erdos (1964) conjectured that a graph with nk nodes each of valence at least ( n - 1) k contains k disjoint complete subgraphs each with n nodes. This and a related conjecture of Grunbaum are discussed and proved in some special cases. Some similar results are obtained showing the existence of disjoint circuits in graphs with sufficiently high valences.