# Unavoidable set of face types for planar maps

@article{Hornk1996UnavoidableSO, title={Unavoidable set of face types for planar maps}, author={Mirko Horn{\'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.

## 26 Citations

On vertex types and cyclic colourings of 3-connected plane graphs

- MathematicsDiscret. Math.
- 2000

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

- MathematicsDiscuss. Math. Graph Theory
- 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

- Mathematics
- 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

- MathematicsSiberian Mathematical Journal
- 2019

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

- Mathematics
- 2017

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

- Mathematics
- 2015

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

- Mathematics
- 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

- MathematicsGraphs Comb.
- 2017

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.

More About the Height of Faces in 3-Polytopes

- MathematicsDiscuss. Math. Graph Theory
- 2018

It is proved that any polytope has a face of degree at most 10 with height at most 20, where 10 and 20 are sharp.

## References

SHOWING 1-10 OF 31 REFERENCES

Simultaneous coloring of edges and faces of plane graphs

- MathematicsDiscret. Math.
- 1994

Joint extension of two theorems of Kotzig on 3-polytopes

- MathematicsComb.
- 1993

It is proved that in each 3-polytope there exists either an edge of weight at most 13 for which both incident faces are triangles, or an edge at most 10 which is incident with a triangle, or else an edge on the other side of these bounds at most 8.

Triangles with restricted degree sum of their boundary vertices in plane graphs

- MathematicsDiscret. Math.
- 1995

Acyclic colorings of planar graphs

- Mathematics
- 1973

A coloring of the vertices of a graph byk colors is called acyclic provided that no circuit is bichromatic. We prove that every planar graph has an acyclic coloring with nine colors, and conjecture…

Cyclic coloration of 3-polytopes

- MathematicsJ. Graph Theory
- 1987

If (G, Phi) is a 3-connected plane graph, then chi sub c p* ( G, Phi)+ 9 is the minimum number of colors in any cyclic coloration of (G), and if rho* is sufficiently large of sufficiently large or sufficiently small, then this bound on chiSub c can be improved somewhat.

Extending Kotzig’s theorem

- Mathematics
- 1983

AbstractThe weight of a graphG is the minimum sum of the two degrees of the end points of edges ofG. Kotzig proved that every graph triangulating the sphere has weight at most 13, and Grünbaum and…

On the total coloring of planar graphs.

- Mathematics
- 1989

The total coloring of a graph G is a coloring of its vertices and edges in which any two adjacent or incident elements of F(G)u£(G) are colored with different colors. Behzad [1] and Vizing [9]…

New Views on Some Old Questions of Combinatorial Geometry

- Mathematics
- 1973

Abstract : Several rather old problems in combinatorial geometry have recently been solved, mostly within the frameworks of either the theory of convex polytopes, or that of arrangements of lines or…