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 polyhedra and clutters associated to A and I respectively. Some applications to Rees algebras and combinatorial optimization are presented. We study a conjecture of Conforti and Cornuéjols using an algebraic approach.

Let I G be the toric ideal of a graph G. We characterize in graph theoretical terms primitive, minimal, indispensable and fundamental binomials of the toric ideal I G .

Let I = I(D) be the edge ideal of a weighted oriented graph D. We determine the irredundant irreducible decomposition of I. Also, we characterize the associated primes and the unmixed property of I. Furthermore, we give a combinatorial characterization for the unmixed property of I, when D is bipartite, D is a whisker or D is a cycle. Finally, we study the Cohen-Macaulay property of I. ⇐) If x ∈ L 3 (C), then by Proposition 2.4, N D (x) ⊆ C \ {x}. Hence, C \ {x} is a vertex cover. A contradiction, since C is minimal. Therefore L 3 (C) = ∅.Definition 2.6. A vertex cover C of D is strong if for each x ∈ L 3 (C) there is (y, x) ∈ E(D) such that y ∈ L 2 (C) ∪ L 3 (C) with y ∈ V + (i.e. w(y) = 1).

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 correspond to ring graphs and that these ideals are minimally generated by Gröbner bases. We prove that any graph can be oriented such that its toric ideal is a complete intersection with a universal Gröbner basis determined by the cycles. It turns out that bipartite ring graphs are exactly the bipartite graphs that have complete intersection toric ideals for any orientation. 0 2000 Mathematics Subject Classification. Primary 05C75; Secondary 05C85, 05C20, 13H10.

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