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A000241
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Crossing number of complete graph with n nodes.
(Formerly M2772 N1115)
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13
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0, 0, 0, 0, 0, 1, 3, 9, 18, 36, 60, 100, 150
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refs;
listen;
history;
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internal format)
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OFFSET
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0,7
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COMMENTS
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It was conjectured by A. Hill in 1958 (see Guy 1960 and Harary-Hill 1963) that a(n) = floor(n/2)*floor((n-1)/2)*floor((n-2)/2)*floor((n-3)/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) 128-134.
Also the sum of the dimensions of the irreducible representations of su(3) that first occur in the (n-5)-th tensor power of the tautological representation. - James Dolan (jdolan(AT)math.ucr.edu), Jun 02 2003
Another version of the conjecture is that a(n)=C(floor(n/2),2)*C(floor((n-1)/2),2).
We conjecture that this sequence is also given by one half of the third coefficient of the denominator polynomial of the n-th 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 3-sphere. 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
a(n) <= A028723(n) for n = 13-21, 23, 25, 27, and 29 based on crossing numbers equal to A028723(n) found using QuickCross. - Eric W. Weisstein, May 02 2019
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REFERENCES
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Ábrego, Bernardo M.; Aichholzer, Oswin; Fernández-Merchant, Silvia; Ramos, Pedro; Salazar, Gelasio. The 2-Page Crossing Number of K_n. Discrete Comput. Geom. 49 (2013), no. 4, 747-777. MR3068573
E. de Klerk, D. V. Pasechnik, and A. Schrijver, "Reduction of Symmetric Semidefinite Programs Using the Regular *-Representation." Math Program. 109 (2007) 613-624.
Jean-Paul 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. 68-72, 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): 333-338.
T. L. Saaty, The number of intersections in complete graphs, Engrg. Cybernetics 9 (1971), no. 6, 1102-1104 (1972).; translated from Izv. Akad. Nauk SSSR Tehn. Kibernet. 1971, no. 6, 151-154 (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. 301-349 of R. L. Graham et al., eds., Handbook of Combinatorics, MIT Press.
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LINKS
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FORMULA
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a(n) ~ n^4/64 (Guy, Kainen).
Empirical g.f.: -x^5*(1+x+x^2)/(x+1)^3/(x-1)^5, which is the same as the conjectured formula of A. Hill. - Simon Plouffe, Feb 06 2013
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CROSSREFS
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It is conjectured that this sequence coincides with A028723.
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KEYWORD
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nonn,more,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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