OFFSET
1,1
COMMENTS
a(n+1) is at most a(n) + 2 since we can add a triangle to a graph with a(n) vertices and increase the number of cycles by 1.
David Eppstein observed that an N-gon with each edge replaced by a triangle has 2^N + N cycles and 2N vertices, and gluing such graphs together greedily yields an upper bound on a(n) of O(log n).
LINKS
David Eppstein, Mathstodon post about O(log n) upper bound, 26 August 2020.
EXAMPLE
For n = 2, a pair of triangles sharing a vertex has five vertices; it is easy to check that no graph on three or four vertices has exactly two cycles, so a(2) = 5.
CROSSREFS
KEYWORD
nonn
AUTHOR
Christopher Purcell, Sep 03 2020
STATUS
approved