Linear Discrepancy and Bandwidth

The linear discrepancy of a partially ordered set P = (X;), denoted by ld(P), is the least integer k for which there exists an injection f : X ! Z satisfying (i) if x y, then f (x) < f (y); and (ii) if x and y are incomparable, then jf(x)?f(y)j k. There is an apparent connection between ld(P) and the bandwidth of the incomparability graph of P. We prove that, in fact, these two quantities are always equal.