A generalized s-star, s ≥ 1, is a tree with a root Z of degree s; all other vertices have degree ≤ 2. 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 of maximum degree ≤ 4(k+i) in G. This bound is tight. Similar results hold for plane graphs.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1216,
author = {Stanislav Jendrol' and Heinz-J\"urgen Voss},
title = {Light classes of generalized stars in polyhedral maps on surfaces},
journal = {Discussiones Mathematicae Graph Theory},
volume = {24},
year = {2004},
pages = {85-107},
zbl = {1059.05039},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1216}
}
Stanislav Jendrol'; Heinz-Jürgen Voss. Light classes of generalized stars in polyhedral maps on surfaces. Discussiones Mathematicae Graph Theory, Tome 24 (2004) pp. 85-107. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1216/
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