On The Number Of Sets Of Cycle Lengths

A set S of integers is called a cycle set on {1, 2, . . .,n} if there exists a graph G on n vertices such that the set of lengths of cycles in G is S. Erdős conjectured that the number of cycle sets on {1, 2, . . .,n} is o(2ⁿ). In this paper, we verify this conjecture by proving that there exists an absolute constant c ≥ 0.1 such that the number of cycle sets on {1, 2, . . .,n} is $o{\left( {2^{{n - n^{c} }} } \right)}$.