Decomposing the cube into paths

We consider the question of when the $n$-dimensional hypercube can be decomposed into paths of length $k$. Mollard and Ramras \cite{MR2013} noted that for odd $n$ it is necessary that $k$ divides $n2^{n-1}$ and that $k\leq n$. Later, Anick and Ramras \cite{AR2013} showed that these two conditions are also sufficient for odd $n \leq 2^{32}$ and conjectured that this was true for all odd $n$. In this note we prove the conjecture.Comment: 7 pages, 2 figure

Decomposing the hypercube Qn into n isomorphic edge-disjoint trees

The original publication is available at www.sciencedirect.comThe pre-print of this article can be found at http://hdl.handle.net/10019.1/16108The problem of finding edge-disjoint trees in a hypercube arises for example in the context of parallel computing [3]. Independently of applications it is of high aesthetic appeal. The hypercube of dimension n, denoted by Qn, comprises 2n vertices each corresponding to a distinct binary string of length n. Two vertices are adjacent if and only if their corresponding binary strings differ in exactly one position. Since each vertex of Qn has degree n, the number of edges is n2n−1. A variety of decomposability options derive from this fact. In the remainder of the introduction we focus on three of them. The first two have been dealt with before in the literature; the third is the topic of this article.Publishers' versio