Long paths and cycles in subgraphs of the cube

TL;DR

This paper proves that subgraphs of the hypercube with high minimum degree necessarily contain exponentially long paths and cycles, with bounds that are shown to be tight, extending to other product graphs.

Contribution

It establishes tight bounds on the length of paths and cycles in subgraphs of the hypercube based on minimum degree, including new results for product-type graphs.

Findings

Subgraphs with minimum degree d contain paths of length 2^d-1.

Such subgraphs contain cycles of length at least 2^d.

Bounds are tight, exemplified by d-dimensional subcubes.

Abstract

Let $Q_n$ denote the graph of the $n$-dimensional cube with vertex set $\{0,1\}^n$ in which two vertices are adjacent if they differ in exactly one coordinate. Suppose $G$ is a subgraph of $Q_n$ with average degree at least $d$. How long a path can we guarantee to find in $G$? Our aim in this paper is to show that $G$ must contain an exponentially long path. In fact, we show that if $G$ has minimum degree at least $d$ then $G$ must contain a path of length $2^d-1$. Note that this bound is tight, as shown by a $d$-dimensional subcube of $Q_n$. We also obtain the slightly stronger result that $G$ must contain a cycle of length at least $2^d$ and prove analogous results for other `product-type' graphs.