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Let I = (x v1 ,. .. , x vq) be a square-free monomial ideal of a polynomial ring K[x 1 ,. .. , x n ] over an arbitrary field K and let A be the incidence matrix with column vectors v 1 ,. .. , v q. We will establish some connections between algebraic properties of certain graded algebras associated to I and combinatorial optimization properties of certain… (More)

- Colin De Verdière, Y De / Gitler, Yves Colin De Verdière, Isidoro Gitler, Dirk Vertigan
- 2011

We study irreducible representations of Rees cones and characterize the max-flow min-cut property of clutters in terms of the normality of Rees algebras and the integrality of certain polyhedra. Then we present some applications to combinatorial optimization and commutative algebra. As a byproduct we obtain an effective method, based on the program Normaliz… (More)

A graph is Y Y-reducible if it can be reduced to a vertex by a sequence of series-parallel reductions and Y Y-transformations. Terminals are distinguished vertices which cannot be deleted by reductions and transformations. In this paper we show that four-terminal planar graphs are Y Y-reducible when at least three of the vertices lie on the same face. Using… (More)

We show that for each integer g ≥ 0 there is a constant c g > 0 such that every graph that embeds in the projective plane with sufficiently large face–width r has crossing number at least c g r 2 in the orientable surface Σ g of genus g. As a corollary, we give a polynomial time constant factor approximation algorithm for the crossing number of projective… (More)

Let G be a graph without isolated vertices and let α(G) be its stability number and τ (G) its covering number. The σ v-cover number of a graph, denoted by σ v (G), is the maximum natural number m such that every vertex of G belongs to a maximal independent set with at least m vertices. In the first part of this paper we prove that α(G) ≤ τ (G)[1 + α(G) − σ… (More)

For a bipartite graph G we are able to characterize the complete intersection property of the edge subring in terms of the multiplicity and we give optimal bounds for this number. We give a method to obtain a regular sequence for the atomic ideal of G, when G is embedded on an orientable surface. We also give a graph theoretical condition for the edge… (More)

A graph is terminal ∆ − Y-reducible if, it can be reduced to a distinguished set of terminal vertices by a sequence of series-parallel reductions and ∆ − Y-transformations. Terminal vertices (o terminals for short) cannot be deleted by reductions and transformations. Reducibility of terminal graphs is very difficult and in general not possible for graphs… (More)

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs… (More)