Abstract :
A hypergraph is 2 -colorable if there is a 2 -coloring of the vertices with no monochromatic hyperedge. Let H k denote the class of all k -uniform k -regular hypergraphs. The Lovász Local Lemma, devised by Erdös and Lovász in 1975 to tackle the problem of hypergraph 2-colorings, implies that every hypergraph H ∈ H k is 2 -colorable, provided k ≥ 9 . Alon and Bregman [N. Alon, Z. Bregman, Every 8-uniform 8-regular hypergraph is 2-colorable, Graphs Combin. 4 (1988) 303–306] proved the slightly stronger result that every hypergraph H ∈ H k is 2 -colorable, provided k ≥ 8 . It is implicitly known in the literature that the Alon–Bregman result is true for all k ≥ 4 , as remarked by Vishwanathan [S. Vishwanathan, On 2-coloring certain k -uniform hypergraphs, J. Combin. Theory Ser. A 101 (2003) 168–172] even though we have not seen it explicitly proved. For completeness, we provide a short proof of this result. As remarked by Alon and Bregman the result is not true when k = 3 , as may be seen by considering the Fano plane.
in result in this paper is a strengthening of the above results. For this purpose, we define a set X of vertices in a hypergraph H to be a free set in H if we can 2 -color V ( H ) ∖ X such that every edge in H receives at least one vertex of each color. Equivalently, X is a free set in H if it is the complement of two disjoint transversals in H . For every k ≥ 13 , we prove that every hypergraph H ∈ H k of order n has a free set of size at least n / 5 . For any ϵ where 0 < ϵ < 1 and for sufficiently large k , we prove that every hypergraph H ∈ H k of order n has a free set of size at least c k n , where c k = 1 − 6 ( 1 + ϵ ) ln ( k ) / k , and so c k → 1 as k → ∞ . As an application, we show that the total restrained domination number of a graph on n vertices with sufficiently large minimum degree k is at most 1 2 ( 1 − c k ) n , which significantly improves the best known bound of 1 2 n + 1 .
