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

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 $V(H) \setminus \{v\}$ 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.