Hypercube geodesics with few colour changes

TL;DR

This paper investigates the minimal number of colour changes in paths within hypercubes under 2-colourings, improving bounds and showing the expected number of changes is approximately A9/2 b7 A0 b7 bdb7 A0b7 A0, which is optimal for random vertices.

Contribution

The authors improve the upper bound on colour changes in hypercube paths and determine the expected number of changes for random start vertices.

Findings

Improved the upper bound on colour changes from B2(n) to (A9/2) b7 A0 b7 bdb7 A0b7 A0.

Showed the expected number of colour changes for a random start vertex is (A9/2) b7 A0 b7 bdb7 A0b7 A0.

Proved this bound is optimal for a uniformly random start vertex.

Abstract

What is the maximum, over all 2-colourings of the edges of the $n$-dimensional hypercube $Q_n$, of the minimal number of times a path between a vertex $v$ and its antipode $\bar{v}$ changes colour? A conjecture of Norine, in a form due to Feder and Subi, states that this maximum should be 1. The previous best-known upper bound on the number of colour changes was $(\tfrac{3}{8} + o(1))n$ due to Dvo\v{r}\'{a}k. We improve this bound and answer a question of Leader and Long by finding a geodesic path with at most $(\tfrac{\pi}{2} + o(1))\sqrt{n}$ colour changes. In fact, we show that this is the expected number of colour changes for a uniformly random start vertex. This is optimal (up to the constant) when the start vertex is chosen uniformly at random.