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## Discussiones Mathematicae Graph Theory

2004 | 24 | 1 | 85-107
Tytuł artykułu

### Light classes of generalized stars in polyhedral maps on surfaces

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Treść / Zawartość
Warianty tytułu
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
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
85-107
Opis fizyczny
Daty
wydano
2004
otrzymano
2002-01-10
poprawiono
2003-05-20
Twórcy
autor
• Department of Geometry and Algebra, P.J. Šafárik University, Jesenná 5, 041 54 Košice, Slovakia
autor
• Department of Algebra, Technical University Dresden, Mommsenstrasse 13, D-01062 Dresden, Germany
Bibliografia
• [1] I. Fabrici and S. Jendrol', Subgraphs with restricted degrees of their vertices in planar 3-connected graphs, Graphs and Combinatorics 13 (1997) 245-250.
• [2] I. Fabrici and S. Jendrol', Subgraphs with restricted degrees of their vertices in planar graphs, Discrete Math. 191 (1998) 83-90, doi: 10.1016/S0012-365X(98)00095-8.
• [3] B. Grünbaum, New views on some old questions of combinatorial geometry (Int. Theorie Combinatorie, Rome, 1973) 1 (1976) 451-468.
• [4] B. Grünbaum and G.C. Shephard, Analogues for tiling of Kotzig's theorem on minimal weight of edges, Ann. Discrete Math. 12 (1982) 129-140.
• [5] J. Harant, S. Jendrol' and M. Tkáč, On 3-connected plane graphs without trianglar faces, J. Combin. Theory (B) 77 (1999) 150-61, doi: 10.1006/jctb.1999.1918.
• [6] J. Ivanco, The weight of a graph, Ann. Discrete Math. 51 (1992) 113-116.
• [7] S. Jendrol', T. Madaras, R. Soták and Zs. Tuza, On light cycles in plane triangulations, Discrete Math. 197/198 (1999) 453-467.
• [8] S. Jendrol' and H.-J. Voss, A local property of polyhedral maps on compact 2-dimensional manifolds, Discrete Math. 212 (2000) 111-120, doi: 10.1016/S0012-365X(99)00329-5.
• [9] S. Jendrol' and H.-J. Voss, A local property of large polyhedral maps on compact 2-dimensional manifolds, Graphs and Combinatorics 15 (1999) 303-313, doi: 10.1007/s003730050064.
• [10] S. Jendrol' and H.-J. Voss, Light paths with an odd number of vertices in large polyhedral maps, Annals of Combin. 2 (1998) 313-324, doi: 10.1007/BF01608528.
• [11] S. Jendrol' and H.-J. Voss, Subgraphs with restricted degrees of their vertices in large polyhedral maps on compact 2-manifolds, European J. Combin. 20 (1999) 821-832, doi: 10.1006/eujc.1999.0341.
• [12] S. Jendrol' and H.-J Voss, Light subgraphs of multigraphs on compact 2-dimensional manifolds, Discrete Math. 233 (2001) 329-351, doi: 10.1016/S0012-365X(00)00250-8.
• [13] S. Jendrol' and H.-J. Voss, Subgraphs with restricted degrees of their vertices in polyhedral maps on compact 2-manifolds, Annals of Combin. 5 (2001) 211-226, doi: 10.1007/PL00001301.
• [14] S. Jendrol' and H.-J. Voss, Light subgraphs of graphs embedded in 2-dimensional manifolds of Euler characteristic ≤ 0 - a survey, in: Paul Erdős and his Mathematics, II (Budapest, 1999) Bolyai Soc. Math. Stud., 11 (János Bolyai Math. Soc., Budapest, 2002) 375-411.
• [15] A. Kotzig, Contribution to the theory of Eulerian polyhedra, Math. Cas. SAV (Math. Slovaca) 5 (1955) 111-113.
• [16] T. Madaras, Note on weights of paths in polyhedral graphs, Discrete Math. 203 (1999) 267-269, doi: 10.1016/S0012-365X(99)00052-7.
• [17] B. Mohar, Face-width of embedded graphs, Math. Slovaca 47 (1997) 35-63.
• [18] G. Ringel, Map color Theorem (Springer-Verlag Berlin, 1974).
• [19] N. Robertson and R. P. Vitray, Representativity of surface embeddings, in: B. Korte, L. Lovász, H.J. Prömel and A. Schrijver, eds., Paths, Flows and VLSI-Layout (Springer-Verlag, Berlin-New York, 1990) 293-328.
• [20] H. Sachs, Einführung in die Theorie der endlichen Graphen, Teil II. (Teubner Leipzig, 1972).
• [21] J. Zaks, Extending Kotzig's theorem, Israel J. Math. 45 (1983) 281-296, doi: 10.1007/BF02804013.
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