F-ideals and f-graphs

ABSTRACTFor a field K, a square-free monomial ideal I of K[x1,…,xn] is called an f-ideal if both its facet complex and Stanley-Reisner complex have the same f-vector. In this paper, we introduce a combinatorial concept (LU-set) and use it to characterize an (n,d)th f-ideal, whose minimal monomial generating set consists of some monomials of a common degree d from K[x1,…,xn]. We classify all (n,2)th f-ideals, thus list all f-graphs whose edge ideals are exactly the (n,2)th f-ideals. Furthermore, we show that all f-graphs are Cohen-Macaulay.