Not-all-equal 3-SAT and 2-colorings of 4-regular 4-uniform hypergraphs |
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Authors: | Michael A Henning Anders Yeo |
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Institution: | 1. Department of Pure and Applied Mathematics, University of Johannesburg, Auckland Park, 2006, South Africa;2. Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark |
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Abstract: | In this paper, we continue our study of 2-colorings in hypergraphs (see, Henning and Yeo, 2013). A hypergraph is 2-colorable if there is a 2-coloring of the vertices with no monochromatic hyperedge. It is known (see Thomassen, 1992) 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 to be a free vertex in if we can 2-color such that every hyperedge in contains vertices of both colors (where has no color). We prove that every 4-uniform 4-regular hypergraph has a free vertex. This proves a conjecture in Henning and Yeo (2015). 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. |
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Keywords: | Hypergraphs Bipartite 2-colorable Transversal Free vertex NAE-3-SAT |
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