Zonal and cozonal labelings using arbitrary abelian groups

Let <span class="math inline">\(G\)</span> be a plane graph with vertex, edge, and region sets <span class="math inline">\(V(G), E(G), F(G)\)</span> respectively. A zonal labeling of a plane graph <span class="math inline">\(G\)</span> is a labeling <span class="math inline">\(\ell: V(G)\rightarrow \{1,2\}\subset \mathbb{Z}_3\)</span> such that for every region <span class="math inline">\(R\in F(G)\)</span> with boundary <span class="math inline">\(B_R\)</span>, <span class="math inline">\(\sum\limits_{v\in V(B_R)}\ell(v)=0\)</span> in <span class="math inline">\(\mathbb{Z}_3\)</span>. We extend this to general abelian groups, defining a <span class="math inline">\(\Gamma\)</span>-zonal labeling as a labeling <span class="math inline">\(\ell:V(G)\rightarrow \Gamma\setminus \{0\}\)</span> such that for every region <span class="math inline">\(R\in F(G)\)</span>, <span class="math inline">\(\sum\limits_{v \in V(B_R)}\ell(v)\)</span> is <span class="math inline">\(0\)</span>. We explore existence of <span class="math inline">\(\Gamma\)</span>-zonal labelings for various families of graphs. We also introduce two variations: generative and strong <span class="math inline">\(\Gamma\)</span>-zonal labelings. A generative <span class="math inline">\(\Gamma\)</span>-zonal labeling is one in which the elements used to label the vertices generate the group <span class="math inline">\(\Gamma\)</span>. A strong <span class="math inline">\(\Gamma\)</span>-zonal labeling is a labeling in which the additive order of <span class="math inline">\(\ell(v)\)</span> is equal to <span class="math inline">\(\deg(v).\)</span> Examples and existence results are provided for both variations. It is shown that strong <span class="math inline">\(\Gamma\)</span>-zonal labelings have a connection to edge colorings that generalizes the connection between zonal labelings and proper edge <span class="math inline">\(3\)</span>-colorings of cubic maps.