Signs of Hamiltonian Circles in Simple Plane Signed Graphs

TL;DR

This paper investigates the signs of Hamiltonian circles in simple plane signed graphs, establishing criteria and local structural conditions for their existence and sign variation.

Contribution

It introduces a face-based approach and co-Hamiltonian sequences to determine the signs of Hamiltonian circles, including new structural theorems for specific configurations.

Findings

A criterion for opposite-sign Hamiltonian circles using co-Hamiltonian sequences.

Structural theorems for ladder and hexagon configurations that realize both signs.

Methods to certify the existence of Hamiltonian circles of both signs without full sequence construction.

Abstract

We study which signs can occur among Hamiltonian circles in simple plane signed graphs. Using a face-based viewpoint, we relate the sign of a Hamiltonian circle to the product of the signs of the faces inside it, and we introduce co-Hamiltonian sequences. This yields a criterion for the existence of opposite-sign Hamiltonian circles via two co-Hamiltonian sequences with opposite face-products. Motivated by signed grid graphs, we develop local structural theorems that allow one to certify the existence of both signs without explicitly constructing the full sequences, including a ladder-type configuration where toggling along two $4$-circles produces Hamiltonian circles of opposite sign, as well as hexagon configurations that realize both signs.