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 Qnd 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 Q2d is simply a torus. In this paper, we prove that if d1≥3 and d2≥3, then Q2d 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.