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We study the Koszul property of a standard graded $K$-algebra $R$ defined bythe binomial edge ideal of a pair of graphs $(G_1,G_2)$. We show that thefollowing statements are equivalent: (i) $R$ is Koszul; (ii) the defining ideal$J_{G_1,G_2}$ of $R$ has a quadratic Gr\"obner basis; (iii) the graded maximalideal of $R$ has linear quotients with respect to a suitable order of itsgenerators

Koszul binomial edge ideals of pairs of graphs