Decomposing the cube into paths

Abstract

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.