The parity Hamiltonian cycle problem

This paper investigates a variant of the Hamiltonian Cycle (HC) problem, named the Parity Hamiltonian Cycle (PHC ) problem: The problem is to find a closed walk visiting each vertex odd number of times, instead of exactly once. We show that the PHC problem is in P even when a closed walk is allowed to use an edge at most z = 4 times, by considering a T -join, which is a generalization of matching. On the other hand, the PHC problem is NP-complete when z = 3. In the case of z = 3 however, the problem is in P when an input graph is four-edge connected, but it still remains NP-complete even when it is two-edge connected. Thus, we are concerned with the hard case in detail, and give a simple necessary and sufficient condition that a two-edge connected C≥5-free (or P6-free) graph has a PHC. Note that the HC problem is known to be NP-complete for those graph classes. This subject is motivated by a new approach to connecting a hard problem HC and an easy problem T -join, by relaxing a constraint of HC.