Koszul binomial edge ideals of pairs of graphs

We study the Koszul property of a standard graded K-algebra R defined by the binomial edge ideal of a pair of graphs (G1,G2). We show that the following statements are equivalent: (i) R is Koszul; (ii) the defining ideal JG1,G2 of R has a quadratic Gröbner basis; (iii) the graded maximal ideal of R has linear quotients with respect to a suitable order of its generators.