Bipartite graphs whose edge algebras are complete intersections

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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) | xᵢ xⱼ ∈ 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 xᵢ xⱼ | (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.