Graph-theoretic analysis demonstrates strong modular connectivity in five-tree planar multigraphs, indicating new dual bounds for nowhere-zero flows and graph homomorphisms.
A graph is called strongly ‐connected if for each boundary function with , there exists an orientation of such that for each . We show that every planar multigraph with 5 edge‐disjoint spanning trees is strongly ‐connected. This verifies a special case of the Additive Base Conjecture when restricted to planar graphs. Hence, every 10‐edge‐connected directed planar graph admits an antisymmetric ‐flow. By duality, every orientation of a planar graph of girth at least 10 admits a homomorphism to a 5‐vertex tournament. Our result also gives a new proof of the known result that every planar graph of girth at least 10 has a homomorphism to the 5‐cycle.
No takes yet. Share an insight, caveat, or question.
Cranston et al. (2026) studied this question.
Synapse has enriched one closely related paper. Consider it for comparative context: