Bilinski diagrams and geodesics in 1-ended planar maps

A Bilinski diagram is a labeling of a planar map with respect to a central vertex and the regional distance of other vertices of the map from that vertex. The class $Ga,b$ consists of all 1-ended, 3-connected planar graphs with the property that every valence is finite and at least a and every covalence is finite and at least b. A map in the class $Ga,b+$ contains no adjacent b-covalent faces, and dually a map in the class $Ga+,b$ contains no adjacent a-valent vertices. It is shown that Bilinski diagrams of maps in $G6,3,G4,4,G3,6,G5,3+$ and $G3+,5$ are uniformly concentric, i.e., the set of vertices at regional distance k from the central vertex induce a circuit for each k ≥ 1. Using this property, an algorithm is developed for constructing geodetic double rays (or geodesics) containing any given edge of a map in $G6,3,G4,4$, or $G5,3+$. A slightly modified algorithm accomplishes the same for maps in $G3,6$. It follows that all Petrie walks in maps in $G3,6$ are geodesics. These results contribute to the known classes of maps satisfying a conjecture of Bonnington, Imrich, and Seifter, without the assumption of vertex-transitivity. In addition, any path in a map in $G6,3,G4,4$, or $G3,6$ that contains at most edges incident with any face or superface (a union of two faces, at least one of which is 3-covalent, with their common incident edge removed) and at most one edge incident with any 3-covalent face is shown to be the unique geodetic path joining its end-vertices. Bilinski diagrams are further utilized to show that the distance sequence of any map in $G6,3,G4,4$, or $G5,3+$ is unimodal.