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Graphs and Combinatorics 4, 303-306 (1988) Combinatorics

Summary: Graphs and Combinatorics 4, 303-306 (1988)
Springer-Verlag 1988
Every 8-Uniform 8-Regular HypergraphIs 2-Colorable
N. Alon* and Z. Bregman
School of Mathematical Sciences,Sackler Faculty of Exact SciencesTel Aviv University,Ramat
Aviv,Tel Aviv,Israel
Abstract. As is wellknown, LovfiszLocal Lemma implies that every d-uniform d-regular hyper-
graph is2-colorable,provided d > 9.Wepresent a differentproofofa slightlystronger result;every
d-uniformd-regular hypergraph is 2-colorable,provided d > 8.
1. Introduction
A d-uniform d-regular hypergraph is a hypergraph in which every edge contains
precisely d vertices and every vertex is contained in precisely d edges. It is well
known (see, e.g., [4], [6]) that for d > 9 each such hypergraph is 2-colorable, i.e.,
there is a 2-coloring of its vertices with no monochromatic edges. To the best of
our knowledge, the only known proof of this fact is the one that applies the Lovhsz
Local Lemma to show that a random vertex coloring of the given hypergraph with
2 colors contains no monochromatic edges with positve (though very small) prob-
ability. Let D be the set of all positive integers d, such that every d-uniform d-regular


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics