Quasi-Hamiltonicity: A Series of Necessary Conditions for a Digraph to Be Hamiltonian

We introduce new necessary conditions, k-quasi-hamiltonicity (0⩽k⩽n−1), for a digraph of order n to be hamiltonian. Every (k+1)-quasi-hamiltonian digraph is also k-quasi-hamiltonian; we construct digraphs which are k-quasi-hamiltonian, but not (k+1)-quasi-hamiltonian. We design an algorithm that checks k-quasi-hamiltonicity of a given digraph with n vertices and m arcs in time O(nmk). We prove that (n−1)-quasi-hamiltonicity coincides with hamiltonicity and 1-quasi-hamiltonicity is equivalent to pseudo-hamiltonicity introduced (for undirected graphs) by L. Babel and G. J. Woeginger (1997, in Lecture Notes in Comput. Sci., Vol. 1335, pp. 38–51, Springer-Verlag, New York/Berlin).