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

Abstract

ABSTRACTLet 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).