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. 
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25 Citations
Background Citations
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25 Citations

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References

SHOWING 1-10 OF 31 REFERENCES
Simultaneous coloring of edges and faces of plane graphs
Joint extension of two theorems of Kotzig on 3-polytopes
TLDR
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
Acyclic colorings of planar graphs
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
TLDR
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
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.
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
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…
Analogues for Tilings of Kotzig'S Theorem on Minimal Weights of Edges*
The Weight of a Graph
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