Long geodesics in subgraphs of the cube

TL;DR

This paper proves that subgraphs of the hypercube with average degree d necessarily contain geodesics of length d, strengthening previous results and connecting to the antipodal colourings conjecture.

Contribution

It establishes a tight lower bound on the length of geodesics in subgraphs of the hypercube based on average degree, improving prior theorems.

Findings

Subgraphs with average degree d contain geodesics of length d.

The result is optimal and generalizes previous theorems.

Links to the antipodal colourings conjecture of Norine.

Abstract

A path in the hypercube $Q_n$ is said to be a geodesic if no two of its edges are in the same direction. Let $G$ be a subgraph of $Q_n$ with average degree $d$. How long a geodesic must $G$ contain? We show that $G$ must contain a geodesic of length $d$. This result, which is best possible, strengthens a theorem of Feder and Subi. It is also related to the `antipodal colourings' conjecture of Norine.