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 edge-union 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 = log2n. We leave open several potentially interesting cases, and raise some other questions of a similar nature. © 2006 Wiley Periodicals, inc.
Journal of Graph Theory
Bollobás, B., & Brightwell, G. (2006). How many graphs are unions of k-cliques?. Journal of Graph Theory, 52 (2), 87-107. https://doi.org/10.1002/jgt.20138