Locating Pairs of Vertices on a Hamiltonian Cycle in Bigraphs

Abstract

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.