How many graphs are unions of k-cliques?

We study the number F[n; k] of n-vertex graphs that can be written as the edgeunion of k-vertex cliques. We obtain reasonably tight estimates for F[n; k] in the cases (i) k = n-o(n) and (ii) k = o(n) but k/ log n . We also show that F[n; k] exhibits a phase transition around k = log2 n. We leave open several potentially interesting cases, and raise some other questions of a similar nature.

• Type: Journal
• Year: 2006
• Where: JGT
• Authors: Béla Bollobás, Graham Brightwell
• Keywords: JGT 2006, K-vertex Cliques, N-vertex Graphs, Tight Estimates