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 \( o{\left( {2^{{n - n^{c} }} } \right)} \).
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Verstraëte, J. On The Number Of Sets Of Cycle Lengths. Combinatorica 24, 719–730 (2004). https://doi.org/10.1007/s00493-004-0043-6
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DOI: https://doi.org/10.1007/s00493-004-0043-6