Light classes of generalized stars in polyhedral maps on surfaces

• Autorzy: Stanislav Jendrol' , Heinz-Jürgen Voss
• Języki publikacji: EN

Abstrakty

EN

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 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 $S_i$ of maximum degree ≤ 4(k+i) in G. This bound is tight. Similar results hold for plane graphs.

Słowa kluczowe

EN

polyhedral maps; embeddings; light subgraphs; path; star; 2-dimensional manifolds; surface

Kategorie tematyczne

• 05C10: Planar graphs; geometric and topological aspects of graph theory
• 05C75: Structural characterization of families of graphs
• 52B10: Three-dimensional polytopes

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2004
• Tom: 24
• Numer: 1
• Strony: 85-107

Daty

wydano

2004

otrzymano

2002-01-10

poprawiono

2003-05-20

Twórcy

autor

Stanislav Jendrol'

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

autor

Heinz-Jürgen Voss

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

Bibliografia

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Identyfikatory

DOI

10.7151/dmgt.1216