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

Abstract

In this paper, 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. It is known (see Thomassen [J. Amer. Math. Soc. 5 (1992), 217-229]) that every 4-uniform 4-regular hypergraph is 2-colorable. Our main result in this paper is a strengthening of this result. For this purpose, we define a vertex in a hypergraph H to be 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). We prove that every 4-uniform 4-regular hypergraph has a free vertex. This proves a known conjecture. Our proofs use a new result on not-all-equal 3-SAT which is also proved in this paper and is of interest in its own right.