Zonal labelings and Tait colorings from a new perspective

Let \(G=(V(G), E(G), F(G))\) $G=(V(G),E(G),F(G))$ 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 \(\ell : V(G)\rightarrow \{1,2\}\subset \mathbb {Z}_3\) $\ell :V(G)\to \{1,2\}\subset {\mathbb{Z}}_3$ such that for every region \(R\in F(G)\) $R\in F(G)$ with boundary \(B_R\) $B_R$ , \(\sum _{v\in V(B_R)}\ell (v)=0\) ${\sum }_{v\in V(B_R)}\ell (v)=0$ in \(\mathbb {Z}_3\) ${\mathbb{Z}}_3$ . 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.