Riassunto
In questo lavoro si introduce un particolare tipo di immersioni cellulari, dette regolari, per grafi colorati sugli spigoli. Facendo uso di tecniche sia geometriche che combinatorie si espongono alcuni teoremi generali di immersione per tali grafi, strettamente collegati si poliedri da essi rappresentati.
Abstract
A particular kind of 2-cell imbeddings, called regular, for edge-coloured graphs is introduced. By using both geometric and combinatorial techniques, some general imbedding theorems for such graphs, strictly related to the polyhedra they represent, are presented.
References
Aleksandrov, P. S.: Topologia Combinatoria. Einaudi, 1957.
Bing, R. H. ‘An Alternative Proof that 3-manifolds can be Triangulated’. Ann. Math. 69 (1959), 37–65.
Cavicchioli, A. and Gagliardi, C.: ‘Crystallization of PL-manifolds with Connected Boundary’. Boll. Un. Mat. Ital. 17B (1980), 902–917.
Cavicchioli, A., Grasselli, L. and Pezzana, M.: ‘Su di una decomposizione normale per le n-varietà chiuse’, Boll. Un Mat. Ital. 17B (1980), 1146–1165.
Ferri, M.: ‘Una rappresentazione delle n-varieta topologiche triangolabili mediante grafi (n+1)-colorati’. Boll. Un. Math. Ital. 13B (1976), 250–260.
Ferri, M.: ‘Crystallisations of 2-fold Branched Coverings of S3’. Proc. Am. Math. Soc. 73 (1979), 271–276.
Fiorini, S.: ‘A Bibliographic Survey of Edge-colorings’. J. Graph Theory 2 (1978), 93–106.
Ferri, M. and Gagliardi, C.: ‘Alcune proprietà caratteristiche delle triangolazioni contratte’. Atti Sem. Mat. Fis. Univ. Modena 24 (1975), 195–220.
Ferri, M. and Gagliardi, C. ‘Cristallisation Moves’. Pacific J. Math. (to appear).
Gagliardi, C.: ‘A combinatorial Characterization of 3-manifold Crystallizations’. Boll. Un. Mat. Ital. 16A (1979), 441–449.
Gagliardi, C.: ‘How to Deduce the Fundamental Group of a Closed n-manifold from a Contracted Triangulation’. J. Comb. Inf. Sys. Sci. 4 (1979), 237–252.
Gagliardi, C.: ‘Extending the Concept of Genus to Dimension n’. Proc. Am. Math. Soc. (to appear).
Glaser, L. C.: Geometrical Combinatorial Topology, Vol. I. Van Nostrand-Reinhold, 1970.
Gross, J. L.: ‘Voltage Graph’, Discrete Math. 9 (1974), 239–246.
Harary, F.: Graph Theory. Addison-Wesley, Reading, 1969.
Heinrich, K.: ‘Unsolved problems’, Summer Research Workship in Algebraic Combinatorics, Simon Fraser University (1979).
Hempel, J.: 3-manifolds, Princeton Univ. Press, 1976.
Hilton, P. J. and Wylie, S.: An Introduction to Algebraic Topology-Homology Theory, Cambridge Univ. Press, 1960.
Lefschetz, S.: Introduction to Topology. Princeton Univ. Press, 1949.
Moise, E. E.: ‘Affine Structures in 3-manifolds V—The Triangulation Theorem and Hauptvermutung’. Ann Math. 56 (1952), 96–114.
Moise, E. E.: Gemoetric Topology in Dimension 2 and 3. Springer-Verlag, 1977.
Pezzana, M.: ‘Sulla struttura topologica delle varietà compatte’. Atti Sem. Mat. Fis. Univ. Modena 23 (1975), 269–277.
Pezzana, M.: ‘Diagrammi di Heegaard e triangolazione contratta’. Boll. Un. Mat. Ital. 12, Suppl. fasc. 3 (1975), 98–105.
Radò, T.: ‘Über der Begriff der Riemannschen Fläche’. Acta Litt. Sci. Szeged 2 (1925), 101–121.
Rourke, C. and Sanderson, B.: Introduction to Piecewise-linear Topology, Springer-Verlag, 1972.
Stahl, S.: ‘Generalized embedding Schemes’. J. Graph Theory 2 (1978), 41–52.
Stahl, S.: ‘The Embeddings of a Graph—A Survey’, J. Graph Theory 2 (1978), 275–298.
Seifert H. and Threlfall, W.: Lehrbuch der Topologie, Teubner, Leipzig, 1934.
Threlfall, W.: ‘Gruppenbilder’, Abh. Math. Phys. Kl. Sächs. Akad. Wiss. 41, n.6 (1932), 1–59.
White, A. T.: Graphs, Groups and Surfaces, North Holland, Amsterdam, 1973.
Youngs, J. W. T.: ‘Minimal Imbeddings and the Genus of Graphs’. J. Math. Mech. 12 (1963), 303–315.
Author information
Authors and Affiliations
Additional information
Work performed under the auspicies of the G.N.S.A.G.A. of the C.N.R. (National Research Council of Italy).
Rights and permissions
About this article
Cite this article
Gagliardi, C. Regular imbeddings of edge-coloured graphs. Geom Dedicata 11, 397–414 (1981). https://doi.org/10.1007/BF00181201
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00181201