Let \(G=(V(G), E(G), F(G))\) be a plane graph with vertex, edge, and region sets V(G), E(G), and F(G) respectively. A zonal labeling of a plane graph G is a labeling \(\ell : V(G)\rightarrow \{1,2\}\subset \mathbb {Z}_3\) such that for every region \(R\in F(G)\) with boundary \(B_R\), \(\sum _{v\in V(B_R)}\ell (v)=0\) in \(\mathbb {Z}_3\). It has been proven by Chartrand, Egan, and Zhang that a cubic map has a zonal labeling if and only if it has a 3-edge coloring, also known as a Tait coloring. A dual notion of cozonal labelings is defined, and an alternate proof of this theorem is given. New features of cozonal labelings and their utility are highlighted along the way. Potential extensions of results to related problems are presented.

设\(G=(V(G), E(G), F(G))\)是一个平面图，其顶点、边和区域集V ( G )、  E ( G ) 和F ( G )分别。平面图G的区域标记是标记\(\ell : V(G)\rightarrow \{1,2\}\subset \mathbb {Z}_3\)使得对于F 中的每个区域 \(R\ (G)\)与边界\(B_R\)，\(\sum _{v\in V(B_R)}\ell (v)=0\)在\(\mathbb {Z}_3\)中。Chartrand、Egan 和Zhang 已经证明，立方图当且仅当它具有 3 边着色（也称为 Tait 着色）时才具有区域标记。定义了共带标记的双重概念，并给出了该定理的替代证明。共区标记的新特征及其实用性在此过程中得到了强调。提出了结果对相关问题的潜在扩展。