The Parity Hamiltonian Cycle Problem in Directed Graphs
This paper investigates a variant of the Hamiltonian cycle, the parity Hamiltonian cycle (PHC) problem: a PHC in a directed graph is a closed walk (possibly using an arc more than once) which visits every vertex odd number of times. Nishiyama et al. (2015) investigated the undirected version of the PHC problem, and gave a simple characterization that a connected undirected graph has a PHC if and only if it has even order or it is non-bipartite. This paper gives a complete characterization when a directed graph has a PHC, and shows that the PHC problem in a directed graph is solved in polynomial time. The characterization, unlike with the undirected case, is described by a linear system over GF(2).
KeywordsHamiltonian cycle T-joins Linear system over GF(2)
- 7.Fiorini, S., Massar, S., Pokutta, S., Tiwary, H.R., de Wolf, R.: Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds. In: Proceedings of STOC 2012, pp. 95–106 (2012)Google Scholar
- 8.Hartvigsen, D.: Extensions of Matching Theory, Ph.D. thesis, Carnegie Mellon University, Pittsburgh, PA (1984)Google Scholar
- 14.Nishiyama, H., Kobayashi, Y., Yamauchi, Y., Kijima, S., Yamashita, M.: The parity Hamiltonian cycle problem. arXiv.org e-Print archive abs/1501.06323Google Scholar