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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
##### Keywords:
number of cycles; cubic graph
Full Text:
##### References:
 [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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