Signs of Hamiltonian Circles in Simple Plane Signed Graphs

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.

Figures (9)

• Figure 1: A plane graph with three interior vertices
• Figure 2: If $O_1$ and $O_2$ share two vertices $v_1,v_2$, then either parallel edges occur (left) or an interior vertex $v_3$ is forced (middle). The rightmost sketch illustrates the decomposition into $G_1,G_2,G_{i_1},G_{i_2}$ around a face $F$.
• Figure 3: 3 by 4 grid with face (box) labels in blue and vertex labels in parentheses
• Figure 4: 3 by 3 grid with labels $(x,y)$ replaced by $(y,x)$
• Figure 5: The resulting graph obtained from a $4\times6$ grid after deleting a co-Hamiltonian sequence.

Theorems & Definitions (32)

• Definition 1: Outer face, outer edges, exterior and interior vertices
• Definition 2: Co-Hamiltonian edge sequence, co-Hamiltonian face sequence, and Hamiltonian set
• Definition 3: Weak dual or face graph
• Definition 4: Outerplane graph
• Definition 5: Outer boundary circle
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3