On 2‐Colorings of Hypergraphs

Abstract

In this article, we continue the study of 2‐colorings in hypergraphs. A hypergraph is 2‐colorable if there is a 2‐coloring of the vertices with no monochromatic hyperedge. Let Hk denote the class of all k‐uniform k‐regular hypergraphs. It is known (see Alon and Bregman [Graphs Combin. 4 (1988) 303–306] and Thomassen [J. Amer. Math. Soc. 5 (1992), 217–229] that every hypergraph H∈scriptHk is 2‐colorable, provided k≥4. As remarked by Alon and Bregman the result is not true when k=3, as may be seen by considering the Fano plane. Indeed there are several constructions for building infinite families of hypergraphs in H3 that are not 2‐colorable. Our main 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. We conjecture that for k≥4, every hypergraph H∈scriptHk has a free set of size k−3 in H. We show that the bound k−3 cannot be improved for any k≥4 and we prove our conjecture when k∈{5,6,7,8}. Our proofs use results from areas such as transversal in hypergraphs, cycles in digraphs, and probabilistic arguments.

Every 4-regular 4-uniform hypergraph has a 2-coloring with a free vertex

“…The conjecture is proved to hold for k ∈ {5, 6, 7, 8}. The case when k = 4 turned out to be more difficult than the cases when k ∈ {5, 6, 7, 8} and was conjectured separately in [4].…”

“…In this paper, we continue the study of 2-colorings in hypergraphs. We adopt the notation and terminology from [3,4]. A hypergraph H = (V, E) is a finite set V = V (H) of elements, called vertices, together with a finite multiset E = E(H) of arbitrary subsets of V , called hyperedges or simply edges.…”

“…The degree of a vertex v in H, denoted d H (v) or simply by d(v) if H is clear from the context, is the number of edges of H which contain v. The hypergraph H is k-regular if every vertex has degree k in H. For k ≥ 2, let H k denote the class of all k-uniform k-regular hypergraphs. The class H k has been widely studied, both in the context of solving problems on total domination as well as in its own right, see for example [1,3,4,5,10].…”

“…A vertex is a free vertex in H if we can 2-color V (H) \ {v} such that every hyperedge in H contains vertices of both colors (where v has no color). In [4] it is conjectured that every hypergraph H ∈ H k , with k ≥ 4, has a free set of size k − 3. Further, if the conjecture is true, then the bound k − 3 cannot be improved for any k ≥ 4, due to the complete k-uniform hypergraph of order k + 1, as such a hypergraph needs two vertices of each color to ensure every edge has vertices of both colors.…”

A note on hypergraph colorings

“…For more results on hypergraph colorings and related problems, we refer the reader to [2,5,6,7,8,9,10,12,13,14,15].…”