Realization and classification of Hamiltonian-circle multisigns

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.