On the 1-switch conjecture

Feder and Subi conjectured that for any 2-coloring of the edges of the n-dimensional cube, there is an antipodal pair of vertices connected by a path that changes color at most once. They proved that if the coloring is such that there are no properly edge colored 4-cycles, the conjecture is true, without a color change. We generalize their theorem by weakening the assumption on the coloring. Our method can be applied to a similar question on any graph, if the condition on the coloring is satisfied. We solve the corresponding problem on the toroidal grid of size 2a×2b.