Monochromatic Paths and a Topological Approach to Norine's Conjecture

Abstract

Motivated by Norine's conjecture, this paper investigates monochromatic antipodal paths in $2$-edge-coloured hypercubes. Our main method relies on a topological criterion applied to triangulated $2$-skeleta. We show that any antipodal colouring of a centrally symmetric, simply connected $2$-complex yields a monochromatic path linking an antipodal pair of vertices. By symmetrically triangulating opposite square faces in certain classes of colourings, we obtain a topological verification of Norine's conjecture for these classes. We also establish quantitative bounds for cases in which only a limited number of square faces present structural obstructions.