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arXiv · 2603.24292

Orientations of $10$-Edge-Connected Planar Multigraphs and Applications

Daniel W. Cranston·Jiaao Li·Bo Su·Zhouningxin Wang·Chunyan Wei
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Abstract

A graph is called strongly $\Z_{2k+1}$-connected if for each boundary function $\beta: V(G)\mapsto \Z_{2k+1}$ with $\sum_{v\in V(G)}\beta(v)\equiv 0\pmod{2k+1}$, there exists an orientation $D$ of $G$ such that $d_D^+(v) - d_D^-(v) \equiv \beta(v) \pmod{2k+1}$ for each $v \in V(G)$. We show that every planar multigraph with $5$ edge-disjoint spanning trees is strongly $\Z_{5}$-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 $\Z_5$-flow. So, 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.

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Daniel W. Cranston, Jiaao Li, Bo Su, Zhouningxin Wang, Chunyan Wei. 2026-03-25. Orientations of $10$-Edge-Connected Planar Multigraphs and Applications. https://arxiv.org/abs/2603.24292

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