

A000241


Crossing number of complete graph with n nodes.
(Formerly M2772 N1115)


11



0, 0, 0, 0, 0, 1, 3, 9, 18, 36, 60, 100, 150, 225, 315, 441, 588
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OFFSET

0,7


COMMENTS

It was conjectured by A. Hill in 1958 (see Guy 1960 and HararyHill 1963) that a(n) = floor(n/2)*floor((n1)/2)*floor((n2)/2)*floor((n3)/2)/4 (see A028723). This is also sometimes referred to as Guy's conjecture.  N. J. A. Sloane, Jan 21 2015
Verified for n = 11, 12 by Shengjun Pan and R. Bruce Richter, in "The Crossing Number of K_11 is 100", J. Graph Theory 56 (2) (2007) 128134.
The values for n >= 13 are only conjectural.
Also the sum of the dimensions of the irreducible representations of su(3) that first occur in the (n5)th tensor power of the tautological representation.  James Dolan (jdolan(AT)math.ucr.edu), Jun 02 2003
From Paul Barry, Oct 02 2008: (Start)
Another version of the conjecture is that a(n)=C(floor(n/2),2)*C(floor((n1)/2),2).
We conjecture that this sequence is also given by one half of the third coefficient of the denominator polynomial of the nth convergent to the g.f. of n!.
(End)
From the Lackenby reference: "One of the most basic questions in knot theory remains unresolved: is crossing number additive under connected sum? In other words, does the equality c(K1#K2) = c(K1) + c(K2) always hold, where c(K) denotes the crossing number of a knot K and K1#K2 is the connected sum of two (oriented) knots K1 and K2? Theorem 1.1. Let K1, . . .,Kn be oriented knots in the 3sphere. Then (c(K1) + . . . + c(Kn)) / 152 <= c(K1# . . . #Kn) <= c(K1) + . . . + c(Kn)."  Jonathan Vos Post, Aug 26 2009
From the Pan and Richter reference: 0.8594 Z(n) <= a(n) <= Z(n), where Z(n) is the conjectured formula (Richter and Thomassen 1997, de Klerk et al. 2007).  Danny Rorabaugh, Mar 12 2015


REFERENCES

Ábrego, Bernardo M.; Aichholzer, Oswin; FernándezMerchant, Silvia; Ramos, Pedro; Salazar, Gelasio. The 2Page Crossing Number of K_n. Discrete Comput. Geom. 49 (2013), no. 4, 747777. MR3068573
E. de Klerk, D. V. Pasechnik, and A. Schrijver, "Reduction of Symmetric Semidefinite Programs Using the Regular *Representation." Math Program. 109 (2007) 613624.
JeanPaul Delahaye, in Pour La Science, Feb. 2013, #424, Logique et Calcul. Le problème de la fabrique de briques. (The problem of the brick factory), in French.
R. K. Guy, The crossing number of the complete graph, Bull. Malayan Math. Soc., Vol. 7, pp. 6872, 1960.
Harary, Frank, and Anthony Hill. "On the number of crossings in a complete graph." Proceedings of the Edinburgh Mathematical Society (Series 2) 13.04 (1963): 333338.
T. L. Saaty, The number of intersections in complete graphs, Engrg. Cybernetics 9 (1971), no. 6, 11021104 (1972).; translated from Izv. Akad. Nauk SSSR Tehn. Kibernet. 1971, no. 6, 151154 (Russian). Math. Rev. 58 #21749.
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).
C. Thomassen, Embeddings and minors, pp. 301349 of R. L. Graham et al., eds., Handbook of Combinatorics, MIT Press.


LINKS

Table of n, a(n) for n=0..16.
Balko, Martin, Radoslav Fulek, and Jan Kynčl. Crossing numbers and combinatorial characterization of monotone drawings of K_n, arXiv preprint arXiv:1312.3679 (2013). Also Discrete Computat. Geom., 53 (2015), 107143.
Drago Bokal, Gasper Fijavz and David R. Wood, The Minor Crossing Number of Graphs with an Excluded Minor, math.CO/0609707.
J. Dolan et al., su(3) and Zarankiewicz's conjecture
P. Erdős and R. K. Guy, Crossing Number Problems The American Mathematical Monthly, Vol. 80, No. 1. (1973), pp. 5258.
Paul C. Kainen, On a problem of P. Erdős, J. Combinatorial Theory 51968 374377. MR0231744 (38 #72)
Marc Lackenby, The crossing number of composite knots, arXiv:0805.4706 [math.GT], 2009. [From Jonathan Vos Post, Aug 26 2009].
D. McQuillan and R. B. Richter, A parity theorem for drawings of complete and bipartite graphs, Amer. Math. Monthly, 117 (2010), 267273.
A. Owens, On the biplanar crossing number, IEEE Trans. Circuit Theory, 18 (1971), 277280.
Shengjun Pan and R. Bruce Richter, The Crossing Number of K_11 is 100, J. Graph Theory 56 (2) (2007) 128134.
R. B. Richter and C. Thomassen, Relations Between Crossing Numbers of Complete and Complete Bipartite Graphs, Amer. Math. Monthly 104 (1997) 131137.
Thomas L. Saaty, On polynomials and crossing numbers of complete graphs, J. Combinatorial Theory Ser. A 10 (1971), 183184. MR0291013 (45 #107)
T. L. Saaty, The Minimum Number Of Intersections In Complete Graphs
Eric Weisstein's World of Mathematics, Graph Crossing Number.
Eric Weisstein's World of Mathematics, Guy's Conjecture.
E. Weisstein, Zarankiewicz's Conjecture.html


FORMULA

a(n) ~ n^4/64 (Guy, Kainen).
Empirical g.f.: x^5*(1+x+x^2)/(x+1)^3/(x1)^5, which is the same as the conjectured formula of A. Hill.  Simon Plouffe, Feb 06 2013


MATHEMATICA

a[n_] := 1/4*Floor[n/2]*Floor[(n1)/2]*Floor[(n2)/2]*Floor[(n3)/2]; Table[a[n], {n, 0, 16}] (* JeanFrançois Alcover, Feb 06 2013, after Hill's conjectured formula *)


CROSSREFS

It is conjectured that A000241 and A028723 agree.
Cf. A007333, A014540, A030179, A121021, A191928.
Sequence in context: A246695 A132920 A127645 * A028723 A213291 A264365
Adjacent sequences: A000238 A000239 A000240 * A000242 A000243 A000244


KEYWORD

nonn,more,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

Bokal et al. link from Jonathan Vos Post, Dec 08 2006
Entry revised by N. J. A. Sloane, Jan 21 2015


STATUS

approved



