Realization and classification of Hamiltonian-circle multisigns

TL;DR

This paper studies the multisigns of Hamiltonian circles in multisigned complete graphs, revealing their structure as subspaces or affine subspaces of \\mathbb{F}_2^m for large graphs, with some exceptions.

Contribution

It characterizes the set of multisigns of Hamiltonian circles in multisigned complete graphs as subspaces or affine subspaces, extending understanding of their algebraic structure.

Findings

Set of multisigns forms a subspace, affine subspace, or entire space \\mathbb{F}_2^m

Results hold for sufficiently large n and fixed m

Identifies exceptional cases where structure differs

Abstract

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