Jan Goedgebeur - cp's OEIS frontend

A307957 Number of planar graphs of order n with exactly one Hamiltonian cycle.

Original entry on oeis.org

0, 0, 1, 2, 3, 12, 49, 460, 4994, 68234, 997486, 15582567, 253005521, 4250680376, 73293572869, 1293638724177

Offset: 1

Jan Goedgebeur, May 08 2019

Jan Goedgebeur, Barbara Meersman, and Carol T. Zamfirescu, Graphs with few Hamiltonian Cycles, arXiv:1812.05650 [math.CO], 2018-2019. See Table 1 at p. 10.
Eric Weisstein's World of Mathematics, Planar Graph.
Eric Weisstein's World of Mathematics, Uniquely Hamiltonian Graph.

Cf. A307956, A374310, A374311.

A307956 Number of graphs of order n with exactly one Hamiltonian cycle.

Original entry on oeis.org

0, 0, 1, 2, 3, 12, 49, 482, 6380, 135252, 3939509, 166800470, 9739584172, 818717312364, 95353226103276

Offset: 1

Jan Goedgebeur, May 08 2019

Jan Goedgebeur, Barbara Meersman, and Carol T. Zamfirescu, Graphs with few Hamiltonian Cycles, arXiv:1812.05650 [math.CO], 2018-2019. See Table 4 at p. 23.
Eric Weisstein's World of Mathematics, Uniquely Hamiltonian Graph.

Cf. A307957, A374312, A374313, A374314, A374315, A374316.

A305354 Number of cubic hypohamiltonian graphs on 2n vertices.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 3, 1, 100, 52, 202, 304

Offset: 1

Jan Goedgebeur, May 31 2018

J. Goedgebeur and C. T. Zamfirescu, Improved bounds for hypohamiltonian graphs, Ars Mathematica Contemporanea, 13(2) (2017), 235-257.
House of Graphs, Hypohamiltonian graphs
Eric Weisstein's World of Mathematics, Cubic Hypohamiltonian Graph

Cf. A218880.

A305351 Number of snarks with circular flow number 5 on 2n vertices.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 9, 25, 98

Offset: 1

Jan Goedgebeur, May 31 2018

J. Goedgebeur, D. Mattiolo and G. Mazzuoccolo, A unified approach to construct snarks with circular flow number 5, arXiv:1804.00957 [math.CO] (2018).
House of Graphs, Snarks
E. Mácajová and A. Raspaud, On the strong circular 5-flow conjecture, J. Graph Theory, 52 (2006), 307-316.

A289576 Number of cyclically 5-connected simple cubic graphs on 2n vertices.

Original entry on oeis.org

1, 2, 9, 47, 440, 5588, 88074, 1572604, 30639576, 641467774

Offset: 5

Jan Goedgebeur, Jul 08 2017

The counts up to 24 vertices were independently verified by Gordon Royle.

G. Brinkmann, J. Goedgebeur and B. D. McKay, Generation of cubic graphs, Discr. Math. Theor. Comp. Sci. 13 (2) (2011) 69-80.

Cf. A175847.

A216834 Number of weak snarks on 2n nodes.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 2, 6, 31, 155, 1297, 12517, 139854, 1764950, 25286953, 404899916

Offset: 1

Jan Goedgebeur, Sep 19 2012

Multiple definitions of snarks exist which vary in strength. Here snarks are cyclically 4-edge connected cubic graphs with chromatic index 4. These are sometimes called weak snarks. Some stronger definitions require snarks to have girth >= 5 or to be cyclically 5-edge connected.

G. Brinkmann, J. Goedgebeur, J. Hagglund, and K. Markstrom, Generation and properties of Snarks, arxiv 1206.6690 [math.CO], 2012-2013.
J. Goedgebeur, E. Máčajová and M. Škoviera, Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44, arXiv:1712.07867 [math.CO], 2017-2019.
House of Graphs, Snarks
Eric Weisstein's World of Mathematics, Weak Snark

Cf. A130315.

a(18) added by Jan Goedgebeur, May 31 2018

A216783 Number of maximal triangle-free graphs with n vertices.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 10, 16, 31, 61, 147, 392, 1274, 5036, 25617, 164796, 1337848, 13734745, 178587364, 2911304940, 58919069858, 1474647067521, 45599075629687

Offset: 1

Jan Goedgebeur, Sep 18 2012

nonn

A maximal triangle-free graph is a triangle-free graph so that the insertion of each new edge introduces a triangle. For graphs of order larger than 2 this is equivalent to being triangle-free and having diameter 2.

S. Brandt, G. Brinkmann and T. Harmuth, The Generation of Maximal Triangle-Free Graphs, Graphs and Combinatorics, 16 (2000), 149-157.

S. Brandt, G. Brinkmann and T. Harmuth, MTF.
Gunnar Brinkmann, Jan Goedgebeur and J.C. Schlage-Puchta, triangleramsey.
Gunnar Brinkmann, Jan Goedgebeur, and Jan-Christoph Schlage-Puchta, Ramsey numbers R(K3,G) for graphs of order 10, arXiv 1208.0501 (2012).
Jan Goedgebeur, On minimal triangle-free 6-chromatic graphs, arXiv:1707.07581 [math.CO] (2017).
House of Graphs, Maximal triangle-free graphs.

Cf. A280020 (labeled graphs).

a(24) added by Jan Goedgebeur, Jun 05 2018

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User: Jan Goedgebeur

A307957 Number of planar graphs of order n with exactly one Hamiltonian cycle.

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A307956 Number of graphs of order n with exactly one Hamiltonian cycle.

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A305354 Number of cubic hypohamiltonian graphs on 2n vertices.

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A305351 Number of snarks with circular flow number 5 on 2n vertices.

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A289576 Number of cyclically 5-connected simple cubic graphs on 2n vertices.

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A216834 Number of weak snarks on 2n nodes.

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A216783 Number of maximal triangle-free graphs with n vertices.

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