Systems with the integer rounding property in normal monomial subrings

Abstract

Let C be a clutter and let A be its incidence matrix. If the linear system x ≥ 0; x A ≤ 1 has the integer rounding property, we give a description of the canonical module and the a-invariant of certain normal subrings associated to C. If the clutter is a connected graph, we describe when the aforementioned linear system has the integer rounding property in combinatorial and algebraic terms using graph theory and the theory of Rees algebras. As a consequence we show that the extended Rees algebra of the edge ideal of a bipartite graph is Gorenstein if and only if the graph is unmixed.

Normality criteria for monomial ideals

“…In particular, we recover the fact that if I is the edge ideal of a connected graph, then the semigroup ring K[NB] is normal if and only if the system x ≥ 0; xA ≤ 1 has the integer rounding property [8,Theorem 3.3].…”

Gorenstein homogeneous subrings of graphs

“…G is called unmixed if every minimal vertex cover has τ (G) elements. A monomial algebra A is Gorenstein if A is Cohen-Macaulay and its canonical module [4]). In this paper we prove that if n is even, S is normal and Gorenstein, then G is bipartite.…”

Combinatorics of symbolic Rees algebras of edge ideals of clutters

“…The study of symbolic powers of edge ideals from the point of view of graph theory and combinatorics was initiated in [28] and further elaborated on in [29,32]. A breakthrough in this area is the translation of combinatorial problems (e.g., the Conforti-Cornuéjols conjecture [6], the max-flow min-cut property, the idealness of a clutter, or the integer rounding property) into algebraic problems of blowup algebras of edge ideals [3,8,10,11,16,17,22].…”