L. Lovasz, Self-dual polytopes and the chromatic number of distance graphs on the sphere, Acta Sci. Math. (Szeged) 45 (1983), no. 1-4, 317-323.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of in nite graphs, one obtains one of the earliest constructions for nite graphs with high chromatic number and high odd girth. Erd os and R. L. Graham conjectured, that the chromatic number is already large if one uses only the single distance instead of the whole H. This was proved by Lov asz [42], who showed that for any p 2(n 1) n 2 (i.e. when exceeds the side length of the regular simplex inscribed the sphere) n (S n 1 ) n 1, and for every n 2 showed the existence of in nitely many s with (S n 1 ) n 1. Lov asz did not use critical con gurations, he ....

L. Lovasz, Self-dual polytopes and the chromatic number of distance graphs on the sphere, Acta Sci. Math. (Szeged) 45 (1983), no. 1-4, 317-323.

....instance for which the ratio is less than .8796 in which the vectors have a nice three dimensional representation. We have also constructed a weighted instance on 103 vertices for which the ratio is less than .8786. These two instances are based on strongly self dual polytopes due to Lov asz [44]. A polytope P in R n is said to be strongly self dual [44] if (i) P is inscribed in the unit sphere, ii) P is circumscribed around the sphere with origin as center and with radius r for some 0 r 1, and (iii) there is a bijection oe between vertices and facets of P such that, for every ....

....the vectors have a nice three dimensional representation. We have also constructed a weighted instance on 103 vertices for which the ratio is less than .8786. These two instances are based on strongly self dual polytopes due to Lov asz [44] A polytope P in R n is said to be strongly self dual [44] if (i) P is inscribed in the unit sphere, ii) P is circumscribed around the sphere with origin as center and with radius r for some 0 r 1, and (iii) there is a bijection oe between vertices and facets of P such that, for every vertex v of P , the facet oe(v) is orthogonal to the vector v. ....

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L. Lov'asz. Self-dual polytopes and the chromatic number of distance graphs on the sphere. Acta Scientiarum Mathematicarum, 45:317--323, 1983.

....an antipodality, then there exists an antipodal map of S k 1 into T . Lemma 14 Let P be the poset consisting of t consecutive levels of a Boolean algebra on n atoms. Then C(P ) is (t Gamma 2) connected. Corollary 6 The neighborhood complex of K n k is (n Gamma 2k Gamma 1) connected. See [33] for another application of theorem 18. 5.2 The Ham Sandwich Theorem Theorem 19 Let A 1 ; A d be measurable sets in IR d with finite measure. Then there exists a halfspace H such that (H A i ) 1 2 (A i ) for all i. See [1] for another combinatorial application of the ....

L. Lov'asz, Self-dual polytopes and the chromatic number of distance graphs on the sphere, Acta Sci. Math. Szeged 45, 317-323.

....absolute worst case. Although the worst case value of Z MC =Z P is not completely settled, we have constructed instances for which E[W ] Z P 0:8786, showing that the analysis of our algorithm is practically tight. These instances are based on strongly self dual polytopes due to Lov asz [31]. A polytope P in R n is said to be strongly self dual [31] if (i) P is inscribed in the unit sphere, ii) P is circumscribed around the sphere with origin as center and with radius r for some 0 r 1, and (iii) there is a bijection oe between vertices and facets of P such that, for every ....

....=Z P is not completely settled, we have constructed instances for which E[W ] Z P 0:8786, showing that the analysis of our algorithm is practically tight. These instances are based on strongly self dual polytopes due to Lov asz [31] A polytope P in R n is said to be strongly self dual [31] if (i) P is inscribed in the unit sphere, ii) P is circumscribed around the sphere with origin as center and with radius r for some 0 r 1, and (iii) there is a bijection oe between vertices and facets of P such that, for every vertex v of P , the facet oe(v) is orthogonal to the vector v. ....

[Article contains additional citation context not shown here]

L. Lov'asz. Self-dual polytopes and the chromatic number of distance graphs on the sphere. Acta Scientiarum Mathematicarum, 45:317--323, 1983.

....U ) o(n) However, their sets U n also have d Un = p 2 o(1) Thus, following the proof of Proposition 2, their construction is not sufficient to exhibit a family of graphs fG n g for which vc(G n ) sd 0 (G n ) c 0 , for any constant c 0 1. Strongly Self Dual Polytopes. Lov asz in [12] introduces the notion of a strongly self dual polytope, which is defined as a polytope P in R d with the following properties. i) The vertices of P all lie on S d Gamma1 . ii) For some 0 r 1, P is circumscribed around the sphere of radius r centered at the origin. iii) There is a ....

L. Lov'asz, "Self-dual polytopes and the chromatic number of distance graphs on the sphere," Acta Sci. Math., 45(1983), pp. 317--323.

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