An Ordering on the Even Discrete Torus

Abstract

The even discrete torus Tn (k1 ,...,kn ) is the graph on $(\Z/k_1\Z) \times \cdots \times (\Z/k_n\Z)$, where each ki is even and x = (x1 ,...,xn ) is joined to y = (y1 ,...,yn ) if for some i we have xi = yipm 1 and xj = yj for all $j\ne i$. The main aim of this paper is to describe an ordering on the even discrete torus whose initial segments give a best possible isoperimetric inequality. This extends the partial solution of Bollobás and Leader [SIAM J. Discrete Math., 3 (1990), pp. 32--37] to a problem posed by Wang and Wang [SIAM J. Appl. Math., 33 (1977), pp. 55--59].