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Proceedings of the American Mathematical Society

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Reductions of $ n$-fold covers

Author: Saul Stahl
Journal: Proc. Amer. Math. Soc. 72 (1978), 422-424
MSC: Primary 05A05; Secondary 05B40, 05C15
MathSciNet review: 507351
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Abstract: Motivated by L. Lovász's recent proof of the Kneser conjecture [3], this paper offers another result which relates topological and graph theoretical concepts. A method for converting n-fold covers to $ (n - 1)$-fold covers is presented. This yields a strengthening of the classical Borsuk, Lusternik and Schnirelmann theorem on covers of spheres. The same conversion also has applications to multicolorings of graphs.

References [Enhancements On Off] (What's this?)

  • [1] K. Borsuk, Drei Sätze über die n-dimensionale euklidische Sphäre, Fund. Math. 20 (1933), 177-190.
  • [2] David Gale, Neighboring vertices on a convex polyhedron, Linear inequalities and related system, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N.J., 1956, pp. 255–263. MR 0085552
  • [3] L. Lovász, Kneser's conjecture, homotopy and Borsuk's theorem, J. Combinatorial Theory Ser. B (to appear).
  • [4] L. Lusternik and L. Schnirelmann, Méthodes topologiques dans les problèmes variationnels, Gosudarstvennoe Izdat., Moscow, 1930; rev. French transl., Actualités Sci. Indust., no. 118, Hermann, Paris, 1934.
  • [5] Saul Stahl, 𝑛-tuple colorings and associated graphs, J. Combinatorial Theory Ser. B 20 (1976), no. 2, 185–203. MR 0406850

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