On the 1-switch conjecture in the Hypercube and other graphs

TL;DR

This paper investigates the 1-switch conjecture in hypercubes and general graphs, proving it for certain colorings and automorphisms, and extends the problem to other graph structures like toroidal grids.

Contribution

It proves the 1-switch conjecture for a broad class of colorings and automorphisms, and generalizes the problem to various graphs beyond hypercubes.

Findings

Proved the conjecture for random colorings.

Extended the problem to general graphs with automorphisms.

Solved the problem for toroidal grids with specific automorphisms.

Abstract

Feder and Subi conjectured that for any $2$-coloring of the edges of the $n$-dimensional cube, we can find an antipodal pair of vertices connected by a path that changes color at most once. We discuss the case of random colorings, and we prove the conjecture for a wide class of colorings. Our method can be applied to a more general problem, where $Q_n$ can be replaced by any graph $G$, the notion of antipodality by a fixed automorphism $\phi \in Aut(G)$. Thus for any $2$-coloring of $E(G)$ we are looking for a pair of vertices $u,v$ such that $u= \phi(v)$ and there is a path between them with as few color changes as possible. We solve this problem for the toroidal grid $G=C_{2a} \square c_{2b}$ with the automorphism that takes every vertex to its unique farthest pair. Our results point towards a more general conjecture which turns out to be supported by a previous theorem of Feder and Subi.