Discussiones Mathematicae Graph Theory 24(1) (2004) 85-107
DOI: https://doi.org/10.7151/dmgt.1216
LIGHT CLASSES OF GENERALIZED STARS IN POLYHEDRAL MAPS ON SURFACES
| Stanislav Jendrol'
Department of Geometry and Algebra |
Department of Algebra, Technical University Dresden |
| 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.
References
| [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. |
Received 10 January 2002
Revised 20 May 2003