The Edge Ideals of Complete Multipartite Hypergraphs

Abstract

Abstract. We classify all complete uniform multipartite hypergraphs with respect to some algebraic properties, such as being (almost) complete intersection, Gorenstein, level, l-Cohen-Macaulay, l-Buchsbaum, unmixed, and satisfying Serre's condition Sr, via some combinatorial terms. Also, we prove that for a complete s-uniform t-partite hypergraph H, vertex decomposability, shellability, sequentially Sr and sequentially Cohen-Macaulay properties coincide with the condition that H has t−1 sides consisting of a single vertex. Moreover, we show that the latter condition occurs if and only if it is a chordal hypergraph. IntroductionIn recent years, many authors have focused on studying different kinds of monomial ideals associated to combinatorial objects, such as Stanley-Reisner ideals, facet ideals, edge ideals and path ideals (see for example [3], [10], [11], [12], [16] and [17]). In this paper, we study edge ideals of hypergraphs.A hypergraph H with finite vertex set V (H) is a family of nonempty subsets of V (H) whose union is V (H), called edges. The set of vertices and edges of H are denoted by V (H) and E(H), respectively. Sometimes, we also use H as its set of edges. An induced subhypergraph of H, over S ⊆ V (H), is defined as H S = {e ∈ H : e ⊆ S}. If all the edges of a hypergraph H have the same cardinality t, then it is said that H is t-uniform (also sometimes referred as a t-graph). We call an edge of cardinality one, an isolated vertex. If none of the edges of H is included in another, then H is called a simple hypergraph. Throughout this paper, we mean by a hypergraph, a simple one.In this paper, we focus on a class of uniform hypergraphs called t-partite which is a generalization of multipartite graphs. These hypergraphs are important from the combinatorial point of view. The organization of this paper is as follows. In Section 2, we bring some definitions and general properties of hypergraphs and simplicial complexes, and also some relations between combinatorial objects and algebraic ones. For this purpose, we mostly use [1] and [5]. Moreover, we introduce the class of multipartite hypergraphs, and in the case of complete multipartite hypergraphs, we pose some of their basic properties, which will be used in the other sections. In Section 3, we investigate about some algebraic properties of complete multipartite hypergraphs, like being (almost) complete intersection, Gorenstein, level, l-Cohen-Macaulay, l-Buchsbaum, unmixed, and satisfying Serre's condition S r , via studying their independent complexes. For a complete s-uniform t-partite hypergraph H, we show that level, Cohen-Macaulay and S r properties are equivalent to the condition that all sides of H have just one vertex, which happens just in the case that the independent complex of H is a matroid. On the other hand, we show that the independent complex of a complete s-uniform t-partite hypergraph is a matroid if and only if it is a tight complex. Moreover, we show that when s > 2, Buchsbaumness is also equivalent 2010 Mathematics Subject Cla...