A054760 Table T(n,k) = order of (n,k)-cage (smallest n-regular graph of girth k), n >= 2, k >= 3, read by antidiagonals.

3, 4, 4, 5, 6, 5, 6, 8, 10, 6, 7, 10, 19, 14, 7, 8, 12, 30, 26, 24, 8, 9, 14, 40, 42, 67, 30, 9, 10, 16, 50, 62

Offset

0,1

References

P. R. Christopher, Degree monotonicity of cages, Graph Theory Notes of New York, 38 (2000), 29-32.

Links

Table of n, a(n) for n=0..31.

Andries E. Brouwer, Cages

M. Daven and C. A. Rodger, (k,g)-cages are 3-connected, Discr. Math., 199 (1999), 207-215.

Geoff Exoo, Regular graphs of given degree and girth

G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).

Gordon Royle, Cubic Cages

Gordon Royle, Cages of higher valency

Pak Ken Wong, Cages-a survey, J. Graph Theory 6 (1982), no. 1, 1-22.

Formula

T(k,g) >= A198300(k,g) with equality if and only if: k = 2 and g >= 3; g = 3 and k >= 2; g = 4 and k >= 2; g = 5 and k = 2, 3, 7 or possibly 57; or g = 6, 8, or 12, and there exists a symmetric generalized g/2-gon of order k - 1. - Jason Kimberley, Jan 01 2013

Example

First eight antidiagonals are:
   3  4  5  6  7  8  9 10
   4  6 10 14 24 30 58
   5  8 19 26 67 80
   6 10 30 42  ?
   7 12 40 62
   8 14 50
   9 16
  10

Crossrefs

Moore lower bound: A198300.

Orders of cages: this sequence (n,k), A000066 (3,n), A037233 (4,n), A218553 (5,n), A218554 (6,n), A218555 (7,n), A191595 (n,5).

Graphs not required to be regular: A006787, A006856.

Sequence in context: A316353 A204002 A198300 * A079107 A205837 A262872

Adjacent sequences: A054757 A054758 A054759 * A054761 A054762 A054763

Keyword

nonn,tabl,nice,hard,more

Author

N. J. A. Sloane, Apr 26 2000

Extensions

Edited by Jason Kimberley, Apr 25 2010, Oct 26 2011, Dec 21 2012, Jan 01 2013

Status

approved