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Graphs with Large Girth Not Embeddable in the Sphere

Published online by Cambridge University Press:  01 November 2007

PIERRE CHARBIT  and
STÉPHAN THOMASSÉ
PIERRE CHARBIT
Affiliation:
LIAFA, Université Paris 7 Denis Diderot, 2 place Jussieu, Case 7014, 75251 Paris Cedex 05, France (e-mail: pierre.charbit@liafa.jussieu.fr)
STÉPHAN THOMASSÉ
Affiliation:
LIRMM, Université Montpellier 2, 161 rue Ada, 34392 Montpellier Cedex 5, France (e-mail: thomasse@lirmm.fr)
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Abstract

In 1972, Rosenfeld asked if every triangle-free graph could be embedded in the unit sphere Sd in such a way that two vertices joined by an edge have distance more than (ie, distance more than 2π/3 on the sphere). In 1978, Larman [LAR] disproved this conjecture, constructing a triangle-free graph for which the minimum length of an edge could not exceed . In addition, he conjectured that the right answer would be , which is not better than the class of all graphs. Larman'sconjecture was independently proved by Rosenfeld [MR] and Rödl [VR[. In this last paper it was shown that no bound better than can be found for graphs with arbitrarily large odd girth. We prove in this paper that this is stilltrue for arbitrarily large girth. We discuss then the case of triangle-free graphs with linear minimum degree.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 16 , Issue 6 , November 2007 , pp. 829 - 832
Copyright
Copyright © Cambridge University Press 2007

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