The Parity Hamiltonian Cycle Problem

TL;DR

This paper studies the parity Hamiltonian cycle problem, providing characterizations, algorithms for specific graph classes, and complexity results, revealing both tractable cases and NP-hardness in general.

Contribution

It offers a complete characterization of graphs with parity Hamiltonian cycles and develops efficient algorithms for certain classes, while establishing NP-hardness in others.

Findings

Linear time algorithm for PHC with edges used at most four times

NP-hardness of PHCz for z=1,2,3 in two-edge connected graphs

Polynomial-time algorithms for PHC3 in four-edge connected graphs and certain sparse classes

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 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.