DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 16(2) (1996) 123-141
DOI: 10.7151/dmgt.1028

UNAVOIDABLE SET OF FACE TYPES FOR PLANAR MAPS

Mirko Horňák  and Stanislav Jendrol

Department of Geometry and Algebra, P. J. Šafárik University
Jesenná 5, 041 54 Košice, Slovakia

e-mail: hornak@turing.upjs.sk, jendrol@Košice.upjs.sk

Abstract

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.

Keywords: normal planar map, plane graph, type of a face, unavoidable set, cyclic chromatic number.

1991 Mathematics Subject Classification: Primary 05C75, Secondary 05C10.

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