Non-Hamiltonian 2-regular Digraphs -- Residues

TL;DR

This paper explores the non-Hamiltonian properties of 2-regular directed graphs by developing a permutation-based framework that provides a necessary and sufficient condition for non-Hamiltonicity.

Contribution

It introduces novel concepts of biconjugates, excluded sets, and residues, linking non-Hamiltonicity to permutation sets in the symmetric group, and establishes a precise criterion for non-Hamiltonicity.

Findings

Established a necessary and sufficient condition for non-Hamiltonicity.

Connected non-Hamiltonicity to permutation sets in the symmetric group.

Extended previous decomposition techniques to a broader class of digraphs.

Abstract

In earlier papers, we showed a decomposition of the arcs of 2-diregular digraphs (2-dds) and used it to prove some conditions for these graphs to be non-Hamiltonian; we then extended this decomposition to a larger class of digraphs and used it to construct infinite families of (strongly) connected non-Hamiltonian 2-dds and provided techniques to establish non-Hamiltonicity in special cases. In the present paper, for a subclass of these graphs, we show connections between non-Hamiltonicity and sets of permutations in the full symmetric group S(n) by introducing the concepts of biconjugates, excluded sets and residues; we then use these concepts to prove a necessary and sufficient condition for non-Hamiltonicity.