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

TL;DR

This paper proves that highly edge-connected planar multigraphs have special orientations related to $ ext{Z}_5$, leading to new flow and homomorphism results for planar graphs with large girth.

Contribution

It establishes that planar multigraphs with five edge-disjoint spanning trees are strongly $ ext{Z}_5$-connected, confirming a case of the Additive Base Conjecture for planar graphs.

Findings

Planar multigraphs with 5 edge-disjoint spanning trees are strongly $ ext{Z}_5$-connected.

Every 10-edge-connected directed planar graph admits an antisymmetric $ ext{Z}_5$-flow.

Planar graphs of girth at least 10 have a homomorphism to the 5-cycle.

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.