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A333717
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a(n) is the minimal number of vertices in a simple graph with exactly n cycles.
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0
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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, 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, 7, 7, 7, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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