摘要
Let△bea simplicial complex on[n].TheNF-complex of△is the simplicial complexδ_(NF)(△)on[n]for which the facet ideal of△is equal to the Stanley-Reisner ideal ofδ_(NF)(△).Furthermore,for each k=2,3,....,we introduce the kth NF-complexδ_(NF)^(k)(△),which is inductively defined asδ_(NF)(k)(△)=δ_(NF)(δ_(NF)^(k-1)(△))by settingδ_(NF)^(1)(△)=δ_(NF)(△).One canδ_(NF)^(0)(△)=△.The NF-number of△is the smallest integer k>0 for whichδ_(NF)^(k)(△)■△.In the present paper we are especilly interested in the NF-number of a finite graph,which can be regraded as a simplicial complex of dimension one.It is shown that the NF-number of the finite graph K_(n)■K_(m)on[n+m],which is the disjoint union of the complete graphs K_(n)on[n]and K_(m)on[m]for n≥2 and m≥2 with(n,m)≠(2,2),is equal to n+m+2.As a corollary,we find that the NF-number of the complete bipartite graph K_(n,m)on[n+m]is also equal to n+m+2.
基金
supported by the Higher Education Commission of Pakistan(No.7515/Punjab/NRPU/R&D/HEC/2017).
