Decompositions of $n$-Cube into $2^mn$-Cycles

TL;DR

This paper proves that hypercubes can be decomposed into specific large cycles, extending known results and partially solving conjectures related to cycle decompositions in hypercubes.

Contribution

It establishes that hypercubes have decompositions into $2^mn$-cycles for $n geq 2^m$, advancing understanding of cycle structures in hypercubes.

Findings

Decomposition of $Q_n$ into $2^mn$-cycles for $n geq 2^m$

Path decompositions of hypercubes derived from cycle decompositions

Partial solutions to conjectures by Ramras and Erde

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 $2 \leq l \leq n.$ In this paper, we prove that $Q_n$ has a decomposition into $2^mn$-cycles for $n \geq 2^m.$ 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.