%I #19 Sep 08 2020 03:24:43
%S 3,5,4,6,8,5,4,6,8,6,6,5,5,6,6,8,7,6,6,6,6,5,6,6,7,7,7,7,7,6,7,6,6,7,
%T 6,6,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,6,6,6,7,7,7,7,6,7,7,7,6,7,7,7,8,8,
%U 7,7,7,8,7,7,7,7,7,7,7,7,7,7,7,7,6,7,7
%N a(n) is the minimal number of vertices in a simple graph with exactly n cycles.
%C 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.
%C 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).
%H David Eppstein, <a href="https://mathstodon.xyz/@11011110/104757243863013508">Mathstodon post about O(log n) upper bound</a>, 26 August 2020.
%e 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.
%K nonn
%O 1,1
%A _Christopher Purcell_, Sep 03 2020