# Unavoidable set of face types for planar maps

@article{Hork1996UnavoidableSO, title={Unavoidable set of face types for planar maps}, author={Mirko Horň{\'a}k and Stanislav Jendrol’}, journal={Discuss. Math. Graph Theory}, year={1996}, volume={16}, pages={123-141} }

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 T of face types is found such that in any normal planar map there is a face with type from T . The set 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.

## 28 Citations

### On the Weight of Minor Faces in Triangle-Free 3-Polytopes

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- 2016

The weight w(f) of a face f in a 3-polytope is the degree-sum of vertices incident with f, and is improved to 20, which is best possible.

### Paths with restricted degrees of their vertices in planar graphs

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- 1999

In this paper it is proved that every 3-connected planar graph contains a path on 3 vertices each of which is of degree at most 15 and a path on 4 vertices each of which has degree at most 23.…

### Low Faces of Restricted Degree in 3-Polytopes

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The degree of a vertex or face in a 3-polytope is the number of incident edges. A k-face is one of degree k, a k−-face has degree at most k. The height of a face is the maximum degree of its incident…

### The height of faces of 3-polytopes

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The height of a face in a 3-polytope is the maximum degree of the incident vertices of the face, and the height of a 3-polytope, h, is the minimum height of its faces. A face is pyramidal if it is…

### Heights of minor faces in triangle-free 3-polytopes

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The height h(f) of a face f in a 3-polytope is the maximum of the degrees of vertices incident with f. A 4-face is pyramidal if it is incident with at least three 3-vertices. We note that in the (3,…

### A general upper bound for the cyclic chromatic number of 3-connected plane graphs

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- 2009

The cyclic chromatic number of a plane graph G is the smallest number χc(G) of colors that can be assigned to vertices of G in such a way that whenever two distinct vertices are incident with a…

### A Steinberg-Like Approach to Describing Faces in 3-Polytopes

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The purposes of the paper are to obtain a description of 3-polytope graphs, defined in terms of forbidding certain sets of cycle-lengths, called “(3,3,\infty )”, where all parameters are best possible.

### A Steinberg-Like Approach to Describing Faces in 3-Polytopes

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It is trivial that every 3-polytope has a face of degree at most 5, called minor. Back in 1940, Lebesgue gave an approximate description of minor faces in 3-polytopes. Borodin (Diskretn Anal Issled…

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