On hypercube labellings and antipodal monochromatic paths

A labelling of the n-dimensional hypercube Hn is a mapping that assigns value 0 or 1 to each edge of Hn. A labelling is antipodal if antipodal edges of Hn get assigned different values. It has been conjectured that if Hn, n ≥ 2, is given a labelling that is antipodal, then there exists a pair of antipodal vertices joined by a monochromatic path. This conjecture has been verified by hand for n ≤ 5. In this paper we verify the conjecture in the case where the labelling is simple in the sense that no square xyzt in Hn has value 0 assigned to xy, zt and value 1 assigned to yz, tx, even if the given labelling is not antipodal. The proof is based on a new property of (not necessarily antipodal) simple labellings of Hn. We also exhibit a large class of simple labellings that thus satisfy the conjecture. Finally we conjecture that even if the given labelling is not antipodal, there is always a path joining antipodal vertices that switches labels at most once, which implies the original conjecture. We establish this new conjecture for Hn, n ≤ 5 as well.