Minimizing the Number of Complete Bipartite Graphs in a Ks-Saturated Graph

A graph is F -saturated if it does not contain F as a subgraph but the addition of any edge creates a copy of F . We prove that for s >= 3 and t >= 2, the minimum number of copies of K-1,(t) in a K-s-saturated graph is Theta (n(t)(/2)). More precise results are obtained in the case of K-1,K-2, where determining the minimum number of K-1,K-2's in a K-3-saturated graph is related to the existence of Moore graphs. We prove that for s >= 4 and t >= 3, an n-vertex K-s-saturated graph must have at least Cn(t)(/5+8/5) copies of K-2,(t), and we give an upper bound of O(n(t)(/2+3/2)). These results answer a question of Chakraborti and Loh on extremal K-s-saturated graphs that minimize the number of copies of a fixed graph H. General estimates on the number of K-a,K-b's are also obtained, but finding an asymptotic formula for the number K-a,K-b's in a K-s-saturated graph remains open.