Decompositions of $n$-Cube into $2^{m}n$-Cycles

Abstract

It is known that the $n$-dimensional hypercube $Q_n,$ for $n$ even, has a decomposition into $k$-cycles for $k=n,2n,$ $2^l$ with $2\le l\le n.$ In this paper, we prove that $Q_n$ has a decomposition into $2^{m}n$-cycles for $n\ge 2^m.$ As an immediate consequence of this result, we get path decompositions of $Q_n$ as well. This gives a partial solution to a conjecture posed by Ramras and also, it solves some special cases of a conjecture due to Erde.