The NF-Number of a Simplicial Complex

Abstract

Let [Formula: see text] be a simplicial complex on [Formula: see text]. The [Formula: see text]-complex of [Formula: see text] is the simplicial complex [Formula: see text] on [Formula: see text] for which the facet ideal of [Formula: see text] is equal to the Stanley–Reisner ideal of [Formula: see text]. Furthermore, for each [Formula: see text], we introduce the [Formula: see text]th [Formula: see text]-complex [Formula: see text], which is inductively defined as [Formula: see text] by setting [Formula: see text]. One can set [Formula: see text]. The [Formula: see text]-number of [Formula: see text] is the smallest integer [Formula: see text] for which [Formula: see text]. In the present paper we are especially interested in the [Formula: see text]-number of a finite graph, which can be regraded as a simplicial complex of dimension one. It is shown that the [Formula: see text]-number of the finite graph [Formula: see text] on [Formula: see text], which is the disjoint union of the complete graphs [Formula: see text] on [Formula: see text] and [Formula: see text] on [Formula: see text] for [Formula: see text] and [Formula: see text] with [Formula: see text], is equal to [Formula: see text]. As a corollary, we find that the [Formula: see text]-number of the complete bipartite graph [Formula: see text] on [Formula: see text] is also equal to [Formula: see text].