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Realization and classification of Hamiltonian-circle multisigns

Xiyong Yan

TL;DR

The paper addresses the realization problem for multisigns of Hamiltonian circles in multisigned complete graphs $Σ_n=(K_n,\sigma,\mathbb{F}_2^m)$. It introduces the multisign $σ(C)$ of a circle and proves a constructive realization result via a covering $C_4$-necklace and the $C_4$-necklace Lemma: when the squares’ multisigns span $\mathbb{F}_2^m$, every $g\in\mathbb{F}_2^m$ is attained by some Hamiltonian circle. It further analyzes cases determined by almost-disjoint triangles and the parity of $n$, showing that the set of multisigns is either all of $\mathbb{F}_2^m$ or an affine/subspace, and identifies exceptional configurations where not all multisigns can be realized. Overall, the work extends Hamiltonian-sign results from single/double to multisigned graphs, providing a framework for realizing and classifying multisigns in large multicoded complete graphs and highlighting the roles of triangle spans and necklace constructions in Hamiltonian circle theory.

Abstract

We investigate the multisigns of Hamiltonian circles in the multisigned complete graph \(Σ_n := (K_n, σ, \mathbb{F}_2^m)\). The \emph{multisign} of a circle \(C\) is defined as the sum \[ σ(C) := \sum_{e \in E(C)} σ(e). \] For a fixed \(m\) and sufficiently large \(n\), we show that the set of multisigns of Hamiltonian circles \[ \{σ(H) : H \text{ is a Hamiltonian circle of } Σ_n)\} \] forms either a subspace, an affine subspace, or the entire space \(\mathbb{F}_2^m\), except in certain exceptional cases.

Realization and classification of Hamiltonian-circle multisigns

TL;DR

The paper addresses the realization problem for multisigns of Hamiltonian circles in multisigned complete graphs . It introduces the multisign of a circle and proves a constructive realization result via a covering -necklace and the -necklace Lemma: when the squares’ multisigns span , every is attained by some Hamiltonian circle. It further analyzes cases determined by almost-disjoint triangles and the parity of , showing that the set of multisigns is either all of or an affine/subspace, and identifies exceptional configurations where not all multisigns can be realized. Overall, the work extends Hamiltonian-sign results from single/double to multisigned graphs, providing a framework for realizing and classifying multisigns in large multicoded complete graphs and highlighting the roles of triangle spans and necklace constructions in Hamiltonian circle theory.

Abstract

We investigate the multisigns of Hamiltonian circles in the multisigned complete graph \(Σ_n := (K_n, σ, \mathbb{F}_2^m)\). The \emph{multisign} of a circle is defined as the sum For a fixed and sufficiently large , we show that the set of multisigns of Hamiltonian circles forms either a subspace, an affine subspace, or the entire space , except in certain exceptional cases.

Paper Structure

This paper contains 4 sections, 5 theorems, 39 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Realization and Classification of Hamiltonian-Circle Multisigns
  4. Acknowledgment

Key Result

Lemma 3.1

Suppose that $\Sigma_n:=(K_n,\sigma,\mathbb{F}_2^m)$ contains a covering diamond necklace consisting of paths $p_1, p_2, \dots, p_l$ and squares (or $C_4$'s) and that the set spans the vector space $\mathbb{F}_2^m$. Then $\Sigma_n$ exhibits all possible multisigns of Hamiltonian circles.

Figures (4)

  • Figure 1: A $C_4$-necklace with squares $D_1, D_2,..., D_t$ when $t=4$.
  • Figure 2: almost-disjoint triangles.
  • Figure 3: Four triangles are not almost-disjoint.
  • Figure 4: The triangular path $P_i$

Theorems & Definitions (10)

  • Lemma 3.1: $C_4$-necklace Lemma
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • proof
  • Lemma 3.5
  • proof