The parity Hamiltonian cycle problem

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

Research output: Contribution to journalArticle

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 (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
Volume341
Issue number3
DOIs
Publication statusPublished - Mar 2018

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Hamiltonians
Hamiltonian circuit
Parity
Walk
Graph in graph theory
NP-complete problem
Closed
Graph Classes
Odd number
Linear-time Algorithm
Connected graph
Vertex of a graph

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

Cite this

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

In: Discrete Mathematics, Vol. 341, No. 3, 03.2018, p. 606-626.

Research output: Contribution to journalArticle

Nishiyama, Hiroshi ; Kobayashi, Yusuke ; Yamauchi, Yukiko ; Kijima, Shuji ; Yamashita, Masafumi. / The parity Hamiltonian cycle problem. In: Discrete Mathematics. 2018 ; Vol. 341, No. 3. pp. 606-626.
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