Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-Hamiltonian 2-regular Digraphs -- Residues

Munagala V. Ramanath

TL;DR

The paper addresses the non-Hamiltonicity of connected $2$-diregular digraphs by developing a permutation-based framework that links open-route structures to residue sets in the symmetric group $S_n$. Central to the approach are residues, biconjugates, and exclusion sets, which yield a necessary-and-sufficient condition for non-Hamiltonicity under graph splicing and enable principled reductions of the AC count. The authors demonstrate the efficacy of these ideas with both theoretical results and extensive computational experiments, isolating a small number of challenging cases and showing that many can be resolved via residue-based reductions. The work advances practical diagnostics for non-Hamiltonicity in low-degree digraphs and provides a pathway toward polynomial-time checks for fixed AC counts, strengthening our understanding of Hamiltonian structure in sparse 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.

Non-Hamiltonian 2-regular Digraphs -- Residues

TL;DR

The paper addresses the non-Hamiltonicity of connected -diregular digraphs by developing a permutation-based framework that links open-route structures to residue sets in the symmetric group . Central to the approach are residues, biconjugates, and exclusion sets, which yield a necessary-and-sufficient condition for non-Hamiltonicity under graph splicing and enable principled reductions of the AC count. The authors demonstrate the efficacy of these ideas with both theoretical results and extensive computational experiments, isolating a small number of challenging cases and showing that many can be resolved via residue-based reductions. The work advances practical diagnostics for non-Hamiltonicity in low-degree digraphs and provides a pathway toward polynomial-time checks for fixed AC counts, strengthening our understanding of Hamiltonian structure in sparse 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.
Paper Structure (5 sections, 18 theorems, 8 equations, 3 figures, 2 tables)

This paper contains 5 sections, 18 theorems, 8 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Definitions
  3. Residues
  4. Reduction
  5. References

Key Result

Proposition 1

Suppose $a = a_1a_2...a_ia_{i+1}...a_n \in S_n$ is a product of $n$ permutations and $b$ is a rotated product $b = a_{i+1}...a_na_1a_2...a_i$; then, $a$ and $b$ are conjugates and have the same cycle type.

Figures (3)

  • Figure 1: Splicing a pair of 2-graphs
  • Figure 2: Connected, clean, odd, non-2-splittable, non-Hamiltonian 2-dd $G_5 \in \mathcal{B}^6_6$ formed by splicing 2-graphs $G_a, G_b \in \mathcal{A}^{6,clean}_3$ with 4 saturated vertices in each.
  • Figure 3: Open 2-graphs in $\mathcal{A}^{6,clean}_2$ with 2 saturated vertices

Theorems & Definitions (35)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • ...and 25 more