The Parity Hamiltonian Cycle Problem in Directed Graphs
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This paper investigates a variant of the Hamiltonian cycle, the parity Hamiltonian cycle (PHC) problem: a PHC in a directed graph is a closed walk (possibly using an arc more than once) which visits every vertex odd number of times. Nishiyama et al. (2015) investigated the undirected version of the PHC problem, and gave a simple characterization that a connected undirected graph has a PHC if and only if it has even order or it is non-bipartite. This paper gives a complete characterization when a directed graph has a PHC, and shows that the PHC problem in a directed graph is solved in polynomial time. The characterization, unlike with the undirected case, is described by a linear system over GF(2).
KeywordsHamiltonian cycle T-joins Linear system over GF(2)
This work is partly supported by JSPS KAKENHI Grant Number 15K15938, 25700002, 15H02666, and Grant-in-Aid for Scientific Research on Innovative Areas MEXT Japan “Exploring the Limits of Computation (ELC).”
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