Signs of Hamiltonian Circles in Simple Plane Signed Graphs

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.