# zbMATH — the first resource for mathematics

Unavoidable set of face types for planar maps. (English) Zbl 0877.05048
Summary: The type of a face $$f$$ of a planar map is a sequence of degrees of vertices of $$f$$ as they are encountered when traversing the boundary of $$f$$. A set $$\mathcal T$$ of face types is found such that in any normal planar map there is a face with type from $$\mathcal T$$. The set $$\mathcal T$$ has four infinite series of types as, in a certain sense, the minimum possible number. An analogous result is applied to obtain new upper bounds for the cyclic chromatic number of 3-connected planar maps.

##### MSC:
 05C75 Structural characterization of families of graphs 05C10 Planar graphs; geometric and topological aspects of graph theory
Full Text: