Realization and classification of Hamiltonian-circle multisigns

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