# A conjecture of Bertino: from copulas to graph theory

### Abstract

In this article we prove a conjecture of Salvatore Bertino which centers on a study of the function

*F*(*C*) = \int_0^{1/2} *V*_*C*([*t*,*t*+1/2]^2) *dt*,

where *C *is a copula. This function, which arises in the study of the index of dissimilarity between two random variables (see Bertino 1977), is the integral of the *C*-volumes of the family of squares of sidelength 1/2 sliding from the lower left to the upper right along the ascending diagonal of the unit square. The conjecture made in 1999 is that the minimum value of *F *among the Bertino copulas is 1/8, which is the value of *F *at the FrĀ“echet lower bound. The conjecture is established by showing that a certain family of shuffles of *M *is dense in the set of Bertino copulas and then solving a problem in weighted graphs, which is equivalent to the conjecture in the restricted case.