# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

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# Discussiones Mathematicae Graph Theory

## LIGHT CLASSES OF GENERALIZED STARS IN POLYHEDRAL MAPS ON SURFACES

Stanislav Jendrol'

Department of Geometry and Algebra
P.J. Safarik University
Jesenna 5, 041 54 Košice, Slovakia
e-mail: jendrol@Košice.upjs.sk

 Heinz-Jürgen  Voss

Department of Algebra, Technical University Dresden
Mommsenstrasse 13, D-01062 Dresden, Germany

 Dedicated to Professor Hanjo Walther on the occasion of his 60th birthday

## Abstract

A generalized s-star, s ≥ 1, is a tree with a root Z of degree s; all other vertices have degree ≤ 2. Si denotes a generalized 3-star, all three maximal paths starting in Z have exactly i+1 vertices (including Z). Let M be a surface of Euler characteristic χ (M) ≤ 0, and m(M): = ⎣ (5+√ {49− 24χ (M)})/2⎦ . We prove:

(1) Let k ≥ 1, d ≥ m(M) be integers. Each polyhedral map G on M with a k-path (on k vertices) contains a k-path of maximum degree ≤ d in G or a generalized s-star T, s ≤ m(M), on d+2− m(M) vertices with root Z, where Z has degree ≤ k·m(M) and the maximum degree of T∖{Z} is ≤ d in G. Similar results are obtained for the plane and for large polyhedral maps on M.

(2) Let k and i be integers with k ≥ 3, 1 ≤ i ≤ [k/2]. If a polyhedral map G on M with a large enough number of vertices contains a k-path then G contains a k-path or a 3-star Si of maximum degree ≤ 4(k+i) in G. This bound is tight. Similar results hold for plane graphs.

Keywords: polyhedral maps, embeddings, light subgraphs, path, star, 2-dimensional manifolds, surface.

2000 Mathematics Subject Classification: 05C10, 05C75, 52B10.

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