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a(n) is the minimal number of vertices in a simple graph with exactly n cycles.
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%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