Bipartite Graphs Whose Edge Algebras Are Complete Intersections

Let R be monomial sub-algebra of k[x1, . . . , xN ] generated by square free monomials of degree two. This paper addresses the following question: when is R a complete intersection? For such a k-algebra we can associate a graph G whose vertices are x1, . . . , xN and whose edges are {(xi, xj)|xixj ∈ R}. Conversely, for any graph G with vertices {x1, . . . , xN} we define the edge algebra associated with G as the sub-algebra of k[x1, . . . , xN ] generated by the monomials {xixj |(xi, xj) is an edge of G} . We denote this monomial algebra by k[G]. This paper describes all bipartite graphs whose edge algebras are complete intersections.