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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: [email protected])
STÉPHAN THOMASSÉ
Affiliation:
LIRMM, Université Montpellier 2, 161 rue Ada, 34392 Montpellier Cedex 5, France (e-mail: [email protected])
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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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References

[1]Brandt, S. and Thomassé, S. Dense triangle-free graphs are four-colorable: A solution to the ErdHős–Simonovits problem. to appear in J. Combin Theory Ser. B.Google Scholar
[2]ErdHős, P. (1979) Problems and results in graph theory and combinatorial analysis. In Graph Theory and Related Topics (Proc. Conf. Waterloo, 1977), Academic Press, New York, pp. 153–163.Google Scholar
[3]Goemans, M. X. and Williamson, D. P. (1995) Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. Assoc. Comput. Mach. 42 1115–1145.CrossRefGoogle Scholar
[4]Karger, D., Motwani, R. and Sudan, M. (1998) Approximate graph coloring by semidefinite programming. J. Assoc. Comput. Mach. 45 246–265.CrossRefGoogle Scholar
[5]Larman, D. G. (no 1978) A triangle-free graph which cannot be -imbedded in any Euclidean unit sphere. J. Combin. Theory Ser. A 24 162–169.CrossRefGoogle Scholar
[6]NeĚsetĚril, J. and Rosenfeld, M. (1997) Embedding graphs in Euclidean spaces: An exploration guided by Paul Erdos. Geombinatorics 6 143–155.Google Scholar
[7]Rödl, V. (1984) On combinatorial properties of spheres in Euclidean spaces. Combinatorica 4 345–349.CrossRefGoogle Scholar
[8]Rosenfeld, M. (1982) Triangle-free graphs that are not -embeddable in Sd. J. Combin. Theory Ser. B 33 191–195.CrossRefGoogle Scholar