# Long path and cycle decompositions of even hypercubes

## TL;DR

This paper advances the understanding of decomposing hypercubes into paths and cycles, proving new cases and nearly resolving Erde's conjecture by showing decompositions for certain cycle lengths and paths up to a linear factor.

## Contribution

It extends known results by proving cycle decompositions for lengths up to 2^{n+1}/n and nearly confirms Erde's conjecture for paths dividing the hypercube's edges.

## Key findings

- Cycles of length up to 2^{n+1}/n decompose Q_n.
- Q_n can be decomposed into paths of length up to 2^{n}/n.
- Nearly resolves Erde's conjecture for path decompositions.

## Abstract

We consider edge decompositions of the $n$-dimensional hypercube $Q_n$ into isomorphic copies of a given graph $H$. While a number of results are known about decomposing $Q_n$ into graphs from various classes, the simplest cases of paths and cycles of a given length are far from being understood. A conjecture of Erde asserts that if $n$ is even, $\ell < 2^n$ and $\ell$ divides the number of edges of $Q_n$, then the path of length $\ell$ decomposes $Q_n$. Tapadia et al.\ proved that any path of length $2^mn$, where $2^m<n$, satisfying these conditions decomposes $Q_n$. Here, we make progress toward resolving Erde's conjecture by showing that cycles of certain lengths up to $2^{n+1}/n$ decompose $Q_n$. As a consequence, we show that $Q_n$ can be decomposed into copies of any path of length at most $2^{n}/n$ dividing the number of edges of $Q_n$, thereby settling Erde's conjecture up to a linear factor.