Paired many-to-many disjoint path covers of hypertori

Let n be a positive integer, and let d = (d1, d2, ..., d(n)) be an n-tuple of integers such that d(i) >= 2 for all i. A hypertorus Q(n)(d) is a simple graph defined on the vertex set {(v(1), v(2), ..., v(n)) : 0 <= v(i) <= d(i) - 1 for all and has edges between u = (u1, u2,, nu) and v = (v(1), v(2), ..., v(n)) if and only if there exists a unique i such that vertical bar u(i)-v(i)vertical bar = 1 or d(i)- 1, and for all j not equal i, u(f)= v(j); a two-dimensional hypertorus Q(2)(d) is simply a torus. In this paper, we prove that if d(1) >= 3 and d(2) >= 3, then Q(2)(d) is balanced paired 2-to-2 disjoint path coverable if both di are even, and is paired 2-to-2 disjoint path coverable otherwise. We also discuss a connection between this result and the popular game Flow Free. Finally, we prove several related results in higher dimensions. (C) 2016 Elsevier B.V. All rights reserved.