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On the number of cycles in 3-connected cubic graphs. (English) Zbl 0918.05068
Let \(f(n)\) denote the minimum number of cycles in a 3-connected cubic graph. The authors show that \(f(n)\) is superpolynomial, by showing that for \(n\) sufficiently large, \(2^{n^{0.17}}<f(n)<2^{n^{0.95}}\). This confirms a conjecture by C. A. Barefoot, L. Clark and R. Entringer [Congr. Nummerantium 53, 49-62 (1986; Zbl 0623.05033)].

MSC:
05C30 Enumeration in graph theory
05C38 Paths and cycles
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[1] Barefoot, C.A.; Clark, L.; Entringer, R., Cubic graphs with the minimum number of cycles, Congr. numer., 53, 49-62, (1986)
[2] Bondy, J.A.; Simonovits, M., Longest cycles in 3-connected cubic graphs, Canad. J. math., 32, 987-992, (1980) · Zbl 0454.05043
[3] Bondy, J.A., Basic graph theory: paths and circuits, Handbook of combinatorics, (1995), North-Holland Amsterdam, p. 3-110 · Zbl 0849.05044
[4] Jackson, W., Longes cycles in 3-connected cubic graphs, J. combin. theory ser., 41, 17-26, (1986) · Zbl 0591.05040
[5] Erdős, P.; Szkeres, G., A combinatorial problem in geometry, Composito math., 2, 463-470, (1935) · Zbl 0012.27010
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