On the sequentially Cohen–Macaulay properties of almost complete multipartite graphs

ABSTRACT

Let $S=K[X_1,\dots ,X_n]$ be the polynomial ring over a field K and G be a simple graph on {1,…,n}. We take the edge ideal I(G)⊆S of G and investigate algebraic properties of the residue class ring S∕I(G). In particular, we are interested in the sequentially Cohen–Macaulay property and the other related ones of S∕I(G). For this purpose, we introduce an almost complete multipartite graph, which is a generalization of complete multipartite graphs and bipartite graphs. The main result gives a necessary and sufficient condition for S∕I(G) to be sequentially Cohen–Macaulay for an almost complete multipartite graph G. As an application, we estimate the Castelnuovo–Mumford regularity of S∕I(G) (reg(G)) by using the induced matching number of G (im(G)). Consequently, we give a new class of graphs such that reg(G) = im(G).

KEYWORDS:

• Bipartite graphs
• complete multipartite graphs
• edge ideals
• sequentially Cohen–Macaulay
• simplicial complexes
• the regularity

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13F55
• 13H10
• Secondary: 05C99

Acknowledgments

The author gratefully acknowledges the many helpful suggestions of Professor Yukio Nakamura during the preparation of this paper.