Weighted graphs with all Hamiltonian cycles of the same length Original Research Article

In this paper we identify large classes of directed and undirected graphs, not necessarily complete, for which every associated distance matrix with the property that all its Hamiltonian cycles are of equal length takes a particularly simple form. This provides new insights into an open question raised by Krynski recently and disproves a claim that such graphs are necessarily strongly Hamiltonian. As a by-product of this result, we have an alternative simple proof of the characterization of such distance matrices for complete directed and undirected graphs.