The Parity Hamiltonian Cycle Problem
Abstract
Motivated by a relaxed notion of the celebrated Hamiltonian cycle, this paper investigates its variant, parity Hamiltonian cycle (PHC): A PHC of a graph is a closed walk which visits every vertex an odd number of times, where we remark that the walk may use an edge more than once. First, we give a complete characterization of the graphs which have PHCs, and give a linear time algorithm to find a PHC, in which every edge appears at most four times, in fact. In contrast, we show that finding a PHC is NPhard if a closed walk is allowed to use each edge at most z times for each z=1,2,3 (PHCz for short), even when a given graph is twoedge connected. We then further investigate the PHC3 problem, and show that the problem is in P when an input graph is fouredge connected. Finally, we are concerned with three (or two)edge connected graphs, and show that the PHC3 is in P for any C_>=5free or P6free graphs. Note that the Hamiltonian cycle problem is known to be NPhard for those graph classes.
 Publication:

arXiv eprints
 Pub Date:
 January 2015
 arXiv:
 arXiv:1501.06323
 Bibcode:
 2015arXiv150106323N
 Keywords:

 Computer Science  Computational Complexity
 EPrint:
 29 pages, 16 figures