# Light classes of generalized stars in polyhedral maps on surfaces

• Volume: 24, Issue: 1, page 85-107
• ISSN: 2083-5892

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## Abstract

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A generalized s-star, s ≥ 1, is a tree with a root Z of degree s; all other vertices have degree ≤ 2. ${S}_{i}$ denotes a generalized 3-star, all three maximal paths starting in Z have exactly i+1 vertices (including Z). Let be a surface of Euler characteristic χ() ≤ 0, and m():= ⎣(5 + √49-24χ( ))/2⎦. We prove: (1) Let k ≥ 1, d ≥ m() be integers. Each polyhedral map G on 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(), on d + 2- m() vertices with root Z, where Z has degree ≤ k·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 .. (2) Let k and i be integers with k ≥ 3, 1 ≤ i ≤ [k/2]. If a polyhedral map G on with a large enough number of vertices contains a k-path then G contains a k-path or a 3-star ${S}_{i}$ of maximum degree ≤ 4(k+i) in G. This bound is tight. Similar results hold for plane graphs.

## How to cite

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Stanislav Jendrol', and Heinz-Jürgen Voss. "Light classes of generalized stars in polyhedral maps on surfaces." Discussiones Mathematicae Graph Theory 24.1 (2004): 85-107. <http://eudml.org/doc/270443>.

@article{StanislavJendrol2004,
abstract = {A generalized s-star, s ≥ 1, is a tree with a root Z of degree s; all other vertices have degree ≤ 2. $S_i$ denotes a generalized 3-star, all three maximal paths starting in Z have exactly i+1 vertices (including Z). Let be a surface of Euler characteristic χ() ≤ 0, and m():= ⎣(5 + √49-24χ( ))/2⎦. We prove: (1) Let k ≥ 1, d ≥ m() be integers. Each polyhedral map G on 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(), on d + 2- m() vertices with root Z, where Z has degree ≤ k·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 .. (2) Let k and i be integers with k ≥ 3, 1 ≤ i ≤ [k/2]. If a polyhedral map G on with a large enough number of vertices contains a k-path then G contains a k-path or a 3-star $S_i$ of maximum degree ≤ 4(k+i) in G. This bound is tight. Similar results hold for plane graphs.},
author = {Stanislav Jendrol', Heinz-Jürgen Voss},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {polyhedral maps; embeddings; light subgraphs; path; star; 2-dimensional manifolds; surface; paths; polyhedral map; plane graphs},
language = {eng},
number = {1},
pages = {85-107},
title = {Light classes of generalized stars in polyhedral maps on surfaces},
url = {http://eudml.org/doc/270443},
volume = {24},
year = {2004},
}

TY - JOUR
AU - Stanislav Jendrol'
AU - Heinz-Jürgen Voss
TI - Light classes of generalized stars in polyhedral maps on surfaces
JO - Discussiones Mathematicae Graph Theory
PY - 2004
VL - 24
IS - 1
SP - 85
EP - 107
AB - A generalized s-star, s ≥ 1, is a tree with a root Z of degree s; all other vertices have degree ≤ 2. $S_i$ denotes a generalized 3-star, all three maximal paths starting in Z have exactly i+1 vertices (including Z). Let be a surface of Euler characteristic χ() ≤ 0, and m():= ⎣(5 + √49-24χ( ))/2⎦. We prove: (1) Let k ≥ 1, d ≥ m() be integers. Each polyhedral map G on 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(), on d + 2- m() vertices with root Z, where Z has degree ≤ k·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 .. (2) Let k and i be integers with k ≥ 3, 1 ≤ i ≤ [k/2]. If a polyhedral map G on with a large enough number of vertices contains a k-path then G contains a k-path or a 3-star $S_i$ of maximum degree ≤ 4(k+i) in G. This bound is tight. Similar results hold for plane graphs.
LA - eng
KW - polyhedral maps; embeddings; light subgraphs; path; star; 2-dimensional manifolds; surface; paths; polyhedral map; plane graphs
UR - http://eudml.org/doc/270443
ER -

## References

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