Zonal Labeling of Graphs

<p>A planar graph is said to be zonal when is possible to label its vertices with the nonzero elements of <em>ℤ</em><sub>3</sub>, in such a way that the sum of the labels of the vertices on the boundary of each zone is 0 in <em>ℤ</em><sub>3</sub>. In this work we present some conditions that guarantee the existence of a zonal labeling for a number of families of graphs such as unicyclic and outerplanar, including the family of bipartite graphs with connectivity at least 2 whose stable sets have the same cardinality; additionally, we prove that when any edge of a zonal graph is subdivided twice, the resulting graph is zonal as well. Furthermore, we prove that the Cartesian product <em>G</em> × <em>P</em><sub>2</sub> is zonal, when <em>G</em> is a tree, a unicyclic graph, or certain variety of outerplanar graphs. Besides these results, we determine the number of different zonal labelings of the cycle <em>C</em><sub>n</sub>.</p>