2020/05/04 by Takayuki Hibi, Hibi, Takayuki, Hasan Mahmood +1
Computer Science · Mathematics · #05E45 #13F55 #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AC #math.CO #msc:05E45 #msc:13F55
paper · pdf · doi:10.48550/arxiv.2005.01247
7 Pages
arxiv created 2020/05/04 · openalex publication_date 2020/05/04 · arxiv updated 2020/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Δ be a simplicial complex on [n]. The NF-complex of Δ is the simplicial complex δNF(Δ) on [n] for which the facet ideal of Δ is equal to the Stanley--Reisner ideal of δNF(Δ). Furthermore, for each k = 2,3,… , we introduce \em kth NF-complex δ(k)NF(Δ) which is inductively defined by δ(k)NF(Δ) = δNF(δ(k-1)NF(Δ)) with setting δ(1)NF(Δ) = δNF(Δ). One can set δ(0)NF(Δ) = Δ. The NF-number of Δ is the smallest integer k > 0 for which δ(k)NF(Δ) ≃ Δ. In the present paper we are especially 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 Kn\coprod Km on [n + m], which is the disjoint union of the complete graphs Kn on [n] and Km on [m], where n ≥ 2 and m ≥ 2 with (n,m) ≠ (2,2), is equal to n + m + 2. Its corollary says that the NF-number of the complete bipartite graph Kn,m on [n+m] is also equal to n + m + 2.