Paired many-to-many disjoint path covers of hypertori

Let n be a positive integer, and let d = (d1, d2, . . . , dn) be an n-tuple of integers such that di ≥ 2 for all i. A hypertorus Q d n is a simple graph defined on the vertex set {(v1, v2, . . . , vn) : 0 ≤ vi ≤ di − 1 for all i}, and has edges between u = (u1, u2, . . . , un) and v = (v1, v2, . . . , vn) if and only if there exists a unique i such that |ui − vi| = 1 or di − 1, and for all j ≠ i, uj = vj; a two-dimensional hypertorus Q d 2 is simply a torus. In this paper, we prove that if d1 ≥ 3 and d2 ≥ 3, then Q d 2 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. © 2016 Elsevier B.V. All rights reserved.