An 8-flow theorem for signed graphs

Abstract

We prove that a signed graph admits a nowhere-zero $8$-flow provided that it is flow-admissible and the underlying graph admits a nowhere-zero $4$-flow. When combined with the 4-color theorem, this implies that every flow-admissible bridgeless planar signed graph admits a nowhere-zero $8$-flow. Our result improves and generalizes previous results of Li et al. (European J. Combin. 108 (2023), 103627), which state that every flow-admissible signed $3$-edge-colorable cubic graph admits a nowhere-zero $10$-flow, and that every flow-admissible signed hamiltonian graph admits a nowhere-zero $8$-flow.

Figures (2)

• Figure 1: $a =5$, $b = c = 3$. The flow values of blues edges, red edges and yellow edges are $(0,1), (1,0), (1,1)$, respectively.
• Figure 2: $a =4$, $b = c = 2$, The flow values of blues edges, red edges and yellow edges are $(0,1), (1,1), (1,0)$, respectively.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• Lemma 3.1
• proof
• Lemma 3.2