While the full conjecture remains open, several partial results have been established. Most notably, it has been proven that the hypercube is 2-rearrangeable, meaning that any permutation can be partitioned into two partial permutations, each of which can be routed by edge-disjoint paths. This result has applications in
time-sharing approaches and optical networks with multiple wavelengths, where virtual doubling of edges can be achieved without physically adding connections. The 2-rearrangeability result also provides a 2-approximation algorithm for the maximum disjoint paths problem on the hypercube, which has applications in
admission control for high-speed networks. A related result shows that when source and target vertices are separated by at least two levels in the hypercube (in terms of
Hamming weight), there exist two edge-disjoint collections of vertex-disjoint paths connecting them. More generally, it has been conjectured that if source and target vertices are separated by r levels, then r such edge-disjoint collections should exist. However, the existence of non-routable 2-1 routing requests in a dimension does not immediately provide a counterexample to Szymanski's conjecture for that dimension. In H_3, there exist exactly two 2-1 routing requests that cannot be routed and are non-equivalent by automorphism. One of these, denoted g_3, can be extended to any dimension n \geq 3 to produce a non-routable 2-1 routing request g_n in H_n. Computer searches have identified approximately a dozen non-routable 2-1 routing requests in H_4, though not all can be extended to higher dimensions. ==Motivation==