The number of graphs with large forbidden subgraphs
In this note, extending some results of Erdö;s, Frankl, Rödl, Alexeev, Bollobás and Thomason, we determine asymptotically the number of graphs which do not contain certain large subgraphs. In particular, we show that if H1,H2,... are graphs with |Hn|=o(logn) and ×(Hn)=rn+1, then the number Sn of graphs of order n not containing Hn satisfies log2Sn=(1-1/rn+o(1))(n2). We also give a similar statement for forbidden induced subgraphs. © 2010 Elsevier Ltd.
European Journal of Combinatorics
Bollobás, B., & Nikiforov, V. (2010). The number of graphs with large forbidden subgraphs. European Journal of Combinatorics, 31 (8), 1964-1968. https://doi.org/10.1016/j.ejc.2010.05.005