### 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 NP-hard if a closed walk is allowed to use each edge at most z times for each z=1,2,3 (PHC_{z} for short), even when a given graph is two-edge-connected. We then further investigate the PHC_{3} problem, and show that the problem is in P when an input graph is four-edge-connected. Finally, we are concerned with three (or two)-edge-connected graphs, and show that the PHC_{3} problem is in P for any C_{≥5}-free or P_{6}-free graphs. Note that the Hamiltonian cycle problem is known to be NP-hard for those graph classes.

Original language | English |
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Pages (from-to) | 606-626 |

Number of pages | 21 |

Journal | Discrete Mathematics |

Volume | 341 |

Issue number | 3 |

DOIs | |

Publication status | Published - Mar 2018 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*,

*341*(3), 606-626. https://doi.org/10.1016/j.disc.2017.10.025

**The parity Hamiltonian cycle problem.** / Nishiyama, Hiroshi; Kobayashi, Yusuke; Yamauchi, Yukiko; Kijima, Shuji; Yamashita, Masafumi.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 341, no. 3, pp. 606-626. https://doi.org/10.1016/j.disc.2017.10.025

}

TY - JOUR

T1 - The parity Hamiltonian cycle problem

AU - Nishiyama, Hiroshi

AU - Kobayashi, Yusuke

AU - Yamauchi, Yukiko

AU - Kijima, Shuji

AU - Yamashita, Masafumi

PY - 2018/3

Y1 - 2018/3

N2 - 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 NP-hard 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 two-edge-connected. We then further investigate the PHC3 problem, and show that the problem is in P when an input graph is four-edge-connected. Finally, we are concerned with three (or two)-edge-connected graphs, and show that the PHC3 problem is in P for any C≥5-free or P6-free graphs. Note that the Hamiltonian cycle problem is known to be NP-hard for those graph classes.

AB - 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 NP-hard 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 two-edge-connected. We then further investigate the PHC3 problem, and show that the problem is in P when an input graph is four-edge-connected. Finally, we are concerned with three (or two)-edge-connected graphs, and show that the PHC3 problem is in P for any C≥5-free or P6-free graphs. Note that the Hamiltonian cycle problem is known to be NP-hard for those graph classes.

UR - http://www.scopus.com/inward/record.url?scp=85035026090&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85035026090&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2017.10.025

DO - 10.1016/j.disc.2017.10.025

M3 - Article

AN - SCOPUS:85035026090

VL - 341

SP - 606

EP - 626

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 3

ER -