A081621 -id:A081621 - cp's OEIS frontend

A000103 Number of n-node triangulations of sphere in which every node has degree >= 4.

0, 0, 1, 1, 2, 5, 12, 34, 130, 525, 2472, 12400, 65619, 357504, 1992985, 11284042, 64719885, 375126827, 2194439398, 12941995397, 76890024027, 459873914230, 2767364341936, 16747182732792

Offset: 4

N. J. A. Sloane

			a(4)=0, a(5)=0 because the tetrahedron and the 5-bipyramid both have vertices of degree 3. a(6)=1 because of the A000109(6)=2 triangulations with 6 nodes (abcdef) the one corresponding to the octahedron (bcde,afec,abfd,acfe,adfb,bedc) has no node of degree 3, whereas the other triangulation (bcdef,afec,abed,ace,adcbf,aeb) has 2 such nodes.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. Bowen and S. Fisk, Generation of triangulations of the sphere, Math. Comp., 21 (1967), 250-252.
R. Bowen and S. Fisk, Generation of triangulations of the sphere [Annotated scanned copy]
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
CombOS - Combinatorial Object Server, generate planar graphs
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
D. A. Holton and B. D. McKay, The smallest non-hamiltonian 3-connected cubic planar graphs have 38 vertices, J. Combinat. Theory B vol 45, iss. 3 (1988) 305-319.
D. A. Holton and B. D. McKay, Erratum, J. Combinat. Theory B vol 47, iss. 2 (1989) 248.
J. Lederberg, Dendral-64, II, Report to NASA, Dec 1965 [Annotated scanned copy]
Thom Sulanke, Generating triangulations of surfaces (surftri), (also subpages).
Eric Weisstein's World of Mathematics, Cubic Polyhedral Graph

Cf. all triangulations: A000109, triangulations with minimum degree 5: A081621.

More terms from Hugo Pfoertner, Mar 24 2003

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm) from the Surftri web site, May 05 2007

A111358 Numbers of planar triangulations with minimum degree 5 and without separating 3- or 4-cycles - that is 3- or 4-cycles where the interior and exterior contain at least one vertex.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 12, 23, 71, 187, 627, 1970, 6833, 23384, 82625, 292164, 1045329, 3750277, 13532724, 48977625, 177919099, 648145255, 2368046117, 8674199554, 31854078139, 117252592450, 432576302286, 1599320144703, 5925181102878

Offset: 12

Gunnar Brinkmann, Nov 07 2005

nonn

A006791 and this sequence are the same sequence. The correspondence is just that these objects are planar duals of each other. But the offset and step are different: if the cubic graph has 2*n vertices, the dual triangulation has n+2 vertices. - Brendan McKay, May 24 2017

Also the number of 5-connected triangulations on n vertices. - Manfred Scheucher, Mar 17 2023

			The icosahedron is the smallest triangulation with minimum degree 5 and it doesn't contain any separating 3- or 4-cycles. Examples can easily be seen as 2D and 3D pictures using the program CaGe cited above.

G. Brinkmann, CaGe.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph. [Cached copy, pdf file only, no active links, with permission]
G. Brinkmann and Brendan D. McKay, Construction of planar triangulations with minimum degree 5 , Disc. Math. vol 301, iss. 2-3 (2005) 147-163.
D. A. Holton and B. D. McKay, The smallest non-hamiltonian 3-connected cubic planar graphs have 38 vertices, J. Combinat. Theory B vol 45, iss. 3 (1988) 305-319.
D. A. Holton and B. D. McKay, Erratum, J. Combinat. Theory B vol 47, iss. 2 (1989) 248.

Cf. A081621, A007894, A006791, A000109, A007021, A361578.

A241094 Triangle read by rows: T(n,i) = number of gracefully labeled graphs with n edges that do not use the label i, 1 <= i <= n-1, n > 1.

Original entry on oeis.org

0, 1, 1, 4, 4, 4, 18, 24, 24, 18, 96, 144, 144, 96, 600, 960, 1080, 1080, 960, 600, 4320, 7200, 8460, 8460, 8460, 7200, 4320, 35280, 60840, 75600, 80640, 80640, 75600, 60480, 35280, 322560, 564480, 725760, 806400, 806400, 806400, 725760, 564480, 322560

Offset: 2

Christian Barrientos and Sarah Minion, Apr 15 2014

A graph with n edges is graceful if its vertices can be labeled with distinct integers in the range 0,1,...,n in such a way that when the edges are labeled with the absolute differences between the labels of their end-vertices, the n edges have the distinct labels 1,2,...,n.

			For n=7 and i=3, g(7,3) = 1080.
For n=7 and i=5, g(7,5) = 960.
Triangle begins:
[n\i]  [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]
[2]     0;
[3]     1,      1;
[4]     4,      4,      4;
[5]    18,     24,     24,     18;
[6]    96,    144,    144,    144,     96;
[7]   600,    960,   1080,   1080,    960,    600;
[8]  4320,   7200,   8640,   8640,   8640,   7200,   4320;
[9] 35280,  60480,  75600,  80640,  80640,  75600,  60480,  35280;
...
- _Bruno Berselli_, Apr 23 2014

C. Barrientos and S. M. Minion, Enumerating families of labeled graphs, J. Integer Seq., 18(2015), article 15.1.7.
J. A. Gallian, A dynamic survey of graph labeling, Elec. J. Combin., (2013), #DS6.
David A. Sheppard, The factorial representation of major balanced labelled graphs, Discrete Math., 15(1976), no. 4, 379-388.

Cf. A001563, A003022, A004137, A005488, A006967, A033472, A081621, A103300, A117747, A212661.

/* As triangle: */ [[i le Floor(n/2) select Factorial(n-2)*(n-1-i)*i else Factorial(n-2)*(n-i)*(i-1): i in [1..n-1]]: n in [2..10]]; // Bruno Berselli, Apr 23 2014

Labeled:=(i,n) piecewise(n<2 or i<1, -infinity, 1 <= i <= floor(n/2), GAMMA(n-1)*(n-1-i)*i, ceil((n+1)/2) <= i <= n-1, GAMMA(n-1)*(n-i)*(i-1), infinity):

n=10; (* This number must be replaced every time in order to produce the different entries of the sequence *)
For[i = 1, i <= Floor[n/2], i++, g[n_,i_]:=(n-2)!*(n-1-i)*i; Print["g(",n,",",i,")=", g[n,i]]]
For[i = Ceiling[(n+1)/2], i <= (n-1), i++, g[n_,i_]:=(n-2)!*(n-i)*(i-1); Print["g(",n,",",i,")=",g[n,i]]]

For n >=2, if 1 <= i <= floor(n/2), g(n,i) = (n-2)!*(n-1-i)*i; if ceiling((n+1)/2) <= i <= n-1, g(n,i) = (n-2)!*(n-i)*(i-1).
# alternative
A241094 := proc(n,i)
    if n <2 or i<1 or i >= n then
        0;
    elif i <= floor(n/2) then
        GAMMA(n-1)*(n-1-i)*i;
    else
        GAMMA(n-1)*(n-i)*(i-1) ;
    fi ;
end proc:
seq(seq(A241094(n,i),i=1..n-1),n=2..12); # R. J. Mathar, Jul 30 2024

A339546 Number of at least 3-connected planar triangulations on n vertices such that the minimum valence of any vertex in the mesh is maximized and the number of vertices with this minimum valence is minimized.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 6, 6, 15, 17, 40, 45, 89, 116, 199, 271

Offset: 4

Hugo Pfoertner, Dec 08 2020

It is conjectured that for a polyhedron with maximum possible volume inscribed in a sphere the stated condition is necessary. All known polyhedra with conjecturally maximum volume satisfy this condition. For n <= 15 the given condition determines a unique mesh topology for each n, which coincides with the proved optimal solutions for n <= 8.

See A081314 for references and links.

Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Hugo Pfoertner, Construction of sequence, table of vertex degrees of triangulations (2020).

Cf. A081314, A081621, A339260, A339261, A339262, A339263.

A111357 Numbers of planar triangulations with minimum degree 5 and without separating 3-cycles - that is 3-cycles where the interior and exterior contain at least one vertex.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 12, 23, 73, 191, 649, 2054, 7209, 24963, 89376, 320133, 1160752, 4218225, 15414908, 56474453, 207586410, 764855802, 2825168619, 10458049611, 38795658003, 144203518881, 537031911877, 2003618333624, 7488436558647

Offset: 12

Gunnar Brinkmann, Nov 07 2005

nonn

			The icosahedron is the smallest triangulation with minimum degree 5 and it doesn't contain any separating triangles. Examples can easily be seen as 2D and 3D pictures using the program CaGe cited above.

G. Brinkmann, CaGe.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
G. Brinkmann and Brendan D. McKay, Construction of planar triangulations with minimum degree 5 , Disc. Math. vol 301, iss. 2-3 (2005) 147-163.

Cf. A081621, A007894.

cp's OEIS Frontend

A000103 Number of n-node triangulations of sphere in which every node has degree >= 4.

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A111358 Numbers of planar triangulations with minimum degree 5 and without separating 3- or 4-cycles - that is 3- or 4-cycles where the interior and exterior contain at least one vertex.

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A241094 Triangle read by rows: T(n,i) = number of gracefully labeled graphs with n edges that do not use the label i, 1 <= i <= n-1, n > 1.

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A339546 Number of at least 3-connected planar triangulations on n vertices such that the minimum valence of any vertex in the mesh is maximized and the number of vertices with this minimum valence is minimized.

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Author

Keywords

Comments

References

Links

Crossrefs

A111357 Numbers of planar triangulations with minimum degree 5 and without separating 3-cycles - that is 3-cycles where the interior and exterior contain at least one vertex.

Views

Author

Keywords

Examples

Links

Crossrefs