In mathematics, **Szymanski's conjecture**, named after Ted H. Szymanski (1989), states that every permutation on the *n*-dimensional doubly directed hypercube graph can be routed with edge-disjoint paths. That is, if the permutation σ matches each vertex *v* to another vertex σ(*v*), then for each *v* there exists a path in the hypercube graph from *v* to σ(*v*) such that no two paths for two different vertices *u* and *v* use the same edge in the same direction.

Through computer experiments it has been verified that the conjecture is true for *n* ≤ 4 (Baudon, Fertin & Havel 2001). Although the conjecture remains open for *n* ≥ 5, in this case there exist permutations that require the use of paths that are not shortest paths in order to be routed (Lubiw 1990).