Light classes of generalized stars in polyhedral maps on surfaces

Jendrol', Stanislav ; Voss, Heinz-Jürgen

Discussiones Mathematicae. Graph Theory, Tome 24 (2004) no. 1, pp. 85-107.

Résumé

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:

Détail

Keywords: polyhedral maps, embeddings, light subgraphs, path, star, 2-dimensional manifolds, surface

Bibliographie

[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.