TY - JOUR

T1 - The parity Hamiltonian cycle problem

AU - Nishiyama, Hiroshi

AU - Kobayashi, Yusuke

AU - Yamauchi, Yukiko

AU - Kijima, Shuji

AU - Yamashita, Masafumi

N1 - Funding Information:
The authors are deeply grateful to the anonymous reviewers for their valuable comments. This work is partly supported by JSPS KAKENHI Grant Numbers 15K15938 , 25700002 , 15H02666 , and Grant-in-Aid for Scientific Research on Innovative Areas MEXT Japan Exploring the Limits of Computation (ELC) Grant Numbers 24106002 , 24106005 .
Publisher Copyright:
© 2017 Elsevier B.V.

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.

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U2 - 10.1016/j.disc.2017.10.025

DO - 10.1016/j.disc.2017.10.025

M3 - Article

AN - SCOPUS:85035026090

SN - 0012-365X

VL - 341

SP - 606

EP - 626

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 3

ER -