Mixed interval hypergraphs

AbstractWe investigate the coloring properties of mixed interval hypergraphs having two families of subsets: the edges and the co-edges. In every edge at least two vertices have different colors. The notion of a co-edge was introduced recently in Voloshin (1993, 1995): in every such a subset at least two vertices have the same color. The upper (lower) chromatic number is defined as a maximum (minimum) number of colors for which there exists a coloring of a mixed hypergraph using all the colors.We find that for colorable mixed interval hypergraph H the lower chromatic number χ(H) ⩽ 2, the upper chromatic number χ(H) = |X|−s(H), where s(H) is introduced as the so-called sieve number. A characterization of uncolorability of a mixed interval hypergraph is found, namely: such a hypergraph is uncolorable if and only if it contains an obviously uncolorable edge.The co-stability number α.√(H) is the maximum cardinality of a subset of vertices which contains no co-edge. A mixed hypergraph H is called co-perfect if χ(H′) = α√(H′) for every subhypergraph H′. Such minimal non-co-perfect hypergraphs as monostars and cycloids Cr2r−1 have been found in Voloshin (1995). A new class of non-co-perfect mixed hypergraphs called covered co-bi-stars is found in this paper. It is shown that mixed interval hypergraphs are coperfect if and only if they do not contain co-monostars and covered co-bi-stars as subhypergraphs.Linear time algorithms for computing lower and upper chromatic numbers and respective colorings for this class of hypergraphs are suggested

Prime splittings of Determinantal Ideals

We consider determinantal ideals, where the generating minors are encoded in a hypergraph. We study when the generating minors form a Gr\"obner basis. In this case, the ideal is radical, and we can describe algebraic and numerical invariants of these ideals in terms of combinatorial data of their hypergraphs, such as the clique decomposition. In particular, we can construct a minimal free resolution as a tensor product of the minimal free resolution of their cliques. For several classes of hypergraphs we find a combinatorial description of the minimal primes in terms of a prime splitting. That is, we write the determinantal ideal as a sum of smaller determinantal ideals such that each minimal prime is a sum of minimal primes of the summands.Comment: Final version to appear in Communications in Algebr