On Alphatrion's Conjecture about Hamiltonian paths in hypercubes

Alphatrion conjectured that it is possible to label the vertices of an $n$-dimensional hypercube with distinct positive integers such that for every Hamiltonian path $a_1,\dots ,a_{2^n},$ we have $a_i+a_{i+1}$ prime for all $i\mathrm{.}$ We prove the conjecture by proving the more general result that a graph $G = (V, E)$ can be labeled with distinct positive integers such that the edge sum for all $e\in E$ is prime if and only if $G$ is bipartite.