The parity Hamiltonian cycle problem

Hiroshi Nishiyama, Yusuke Kobayashi, Yukiko Yamauchi, Shuji Kijima, Masafumi Yamashita

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)606-626
Number of pages21
JournalDiscrete Mathematics
Issue number3
Publication statusPublished - Mar 2018

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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