Locating Pairs of Vertices on a Hamiltonian Cycle in Bigraphs

Let G be a simple m × m bipartite graph with minimum degree 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 ≤ k < m / 6 is an integer having appropriate parity. We conjecture that this is also true up to k ≤ m .