文摘
Let nn be a positive integer, and let d=(d1,d2,…,dn) be an nn-tuple of integers such that di≥2di≥2 for all ii. A hypertorus Qnd is a simple graph defined on the vertex set {(v1,v2,…,vn):0≤vi≤di−1 for all i}{(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 ii such that |ui−vi|=1|ui−vi|=1 or di−1di−1, and for all j≠ij≠i, uj=vjuj=vj; a two-dimensional hypertorus Q2d is simply a torus. In this paper, we prove that if d1≥3d1≥3 and d2≥3d2≥3, then Q2d is balanced paired 22-to-22 disjoint path coverable if both didi are even, and is paired 22-to-22 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.