Bipartite graphs whose edge algebras are complete intersections

Abstract

Let R be monomial sub-algebra of $k[x_1,...,x_N]$ 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 $x_1,...,x_N$ and whose edges are $\{(x_i,x_j)|x_i{x}_j\in R\}$. Conversely, for any graph G with vertices $\{x_1,...,x_N\}$ we define the {\it edge algebra associated with G} as the sub-algebra of $k[x_1,...,x_N]$ generated by the monomials $x_i{x}_j|(x_i,x_j)\text{is an edge of}G$. We denote this monomial algebra by k[G]. This paper describes all bipartite graphs whose edge algebras are complete intersections.