Bipartite Graphs Whose Edge Algebras Are Complete Intersections

Abstract

Let R be a monomial subalgebra 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 subalgebra 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.