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

Abstract

A graph is called strongly $\Z_{2k+1}$-connected if for each boundary function $β: V(G)\mapsto \Z_{2k+1}$ with $\sum_{v\in V(G)}β(v)\equiv 0\pmod{2k+1}$, there exists an orientation $D$ of $G$ such that $d_D^+(v) - d_D^-(v) \equiv β(v) \pmod{2k+1}$ 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.

Figures (9)

• Figure 1: The 4 small graphs $aK_2$, $T_{a,b,c}$, $W_1$, and $W_2$
• Figure 2: $G_1\in\{W_1, W_2\}$.
• Figure 3: The configuration $T_{1,1,3}$ and related reconstructions of $G$
• Figure 4: All the elements of $\mathcal{F}_1$
• Figure 5: The graphs $G'/\mathcal{P}'$ and $G$.

Theorems & Definitions (61)

• Conjecture 1.1: Planar Circular Flow Conjecture CL20
• Theorem 1.2: CL20DP17
• Theorem 1.3
• Definition 1.4: LLLMMSZ14
• Theorem 1.5
• Theorem 1.6
• Definition 1.7
• Definition 1.8
• proof
• Proposition 2.2