Koszul bipartite graphs (Q1277205)

Let \(R = K[t_1, \ldots , t_d]\) be the polynomial ring in \(d\) indeterminates over a field \(K\). If \(G\) is a bipartite graph on the vertex set \(\{ 1, \ldots , d \}\), define \(K[G]\) to be the subalgebra of \(R\) generated by all monomials \(t_i t_j\) such that \(\{ i,j \}\) is an edge of \(G\). It is shown that if every \(n\)-cycle \((n \geq 6)\) has a chord, then \(K[G]\) is Koszul.