Cozonality in Plane Graphs of Maximum Degree Four

Let G be a plane graph.A cozonal labeling of G is a face labeling using the labels {1, 2} such that for each vertex v, the labels of regions having v on their boundary sum to 0 (mod 3).Cozonal labelings in connected graphs with maximum degree three have been characterized.In this paper, we define a nonconsecutive walk as one where no two consecutive edges along the walk are on the boundary of the same region, and where every internal vertex of the walk is incident with exactly four regions.By considering these walks, cozonal labelings of graphs with maximum degree four are characterized.