The Edge Ideals of Complete Multipartite Hypergraphs

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 S _r , via some combinatorial terms. Also, we prove that for a complete s-uniform t-partite hypergraph ℋ, vertex decomposability, shellability, sequentially S _r , 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.

Key Words:

• Complete s-uniform t-partite hypergraphs
• Edge ideals
• (Sequentially) Cohen–Macaulay
• Vertex decomposable

2010 Mathematics Subject Classification:

• 05E45
• 13F55
• 05C65
• 13H10

ACKNOWLEDGMENTS

We would like to thank Russ Woodroofe who motivated us to look at the vertex decomposability and chordalness in Theorem 4.1. We would also like to thank the referee for his or her useful and valuable comments.

Notes

Communicated by S. Goto.