A Note on Norine's Antipodal-Colouring Conjecture

TL;DR

This paper improves the upper bound on the number of colour changes needed to connect opposite vertices in a 2-coloured discrete cube, advancing understanding of Norine's antipodal-colouring conjecture.

Contribution

It provides a tighter upper bound of approximately 3/8 n colour changes, improving upon the previous bound of n/2.

Findings

Upper bound on colour changes reduced to (3/8 + o(1))n

Progress towards proving Norine's antipodal-colouring conjecture

Enhanced understanding of colour-changing paths in coloured hypercubes

Abstract

Norine's antipodal-colouring conjecture, in a form given by Feder and Subi, asserts that whenever the edges of the discrete cube are 2-coloured there must exist a path between two opposite vertices along which there is at most one colour change. The best bound to date was that there must exist such a path with at most $n/2$ colour changes. Our aim in this note is to improve this upper bound to $(\frac{3}{8}+o(1))n$.