Locating Pairs of Vertices on a Hamiltonian Cycle in Bigraphs

Let G be a simple $$m\times m$$m×m bipartite graph with minimum degree $$\delta (G)\ge m/2+1$$δ(G)≥m/2+1. We prove that for every pair of vertices x, y, there is a Hamiltonian cycle in G such that the distance between x and y along that cycle equals k, where $$2\le k<m/6$$2≤k<m/6 is an integer having appropriate parity. We conjecture that this is also true up to $$k\le m$$k≤m.