Use an optimal Hamiltonian container bound to improve the global estimate

Develop a method that uses an optimal bound of 2^{-\Omega(\sqrt{p})} in Lemma \ref{lemma:count_ham} to obtain a global upper bound of 2^{n-\Omega(\sqrt{n})} for the number of distinct cycle sets of n-vertex graphs.

Background

Theorem 1 loses a logarithmic factor in the exponent, yielding an upper bound of 2^{n-\Omega(\sqrt{n}/\log^{3/2} n)}. The authors explain that even removing the logarithmic factor from the container estimate for Hamiltonian graphs would not by itself immediately produce the desired global bound.

The main obstruction is the family of graphs with a long cycle and only a linear number of edges, denoted G_2. Raising the edge threshold for the denser family G_4 would make the container argument more effective, but would leave G_2 too large for the elementary counting argument used in the paper.