Skip to main content
SpringerLink
Log in

Sulla tracciabilita' di grafi finiti su superficie compatte

  • Conferenze
  • Published:
Rendiconti del Seminario Matematico e Fisico di Milano Aims and scope Submit manuscript

Sunto

Si illustra il problema della tracciabilità di un grafo finito su superficie compatte e si espongono alcuni tra i risultati più significativi.

Summary

This is an expository paper about the imbedding problem of a finite graph in compact surfaces. Some of the most interesting results are also told.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Bibliografia

  1. Berge C.,Théorie des graphes et ses applications: Dunod, Paris, 1958.

    Google Scholar 

  2. Harary E.,Graph Theory. Addison-Wesley, Reading Mass. 1969.

    Google Scholar 

  3. König, D.,Theorie der endlichen und unendlichen Graphen. Akad. Verlagsgesellschaft, Leipzig, 1936.

    Google Scholar 

  4. Ore O.,The Four-color Problem. Academic Press, New York 1967.

    MATH  Google Scholar 

  5. Ringel G.,Färbungsprobleme auf Flächen und Graphen. Berlin VEB Deutscher Verlag der Wiss. 1959.

    MATH  Google Scholar 

  6. Ringel G.,Map Color Theorem. Springer-Verlag Berlin 1974.

    MATH  Google Scholar 

  7. Seifert H.—Threlfall W.,Lehrbuch der Topologie. Teubner, Leipzig-Berlin, 1934.

    Google Scholar 

  8. White A.,Graphs Groups and Surfaces. North-Holland Publ. Comp. Amsterdam 1973.

    MATH  Google Scholar 

  9. Andreatta A.,Sistemi di circuiti tracciati su una superficie omeomorfa al toro e loro modelli algebrici. Ann. Mat. Pura App. (4), 56, 1961, pp. 99–130.

    Article  MATH  MathSciNet  Google Scholar 

  10. Andreatta A.,Alcuni sviluppi sulla teoria relativa dei singrammi finiti. Ann. Mat. Pura Appl. (4) 65, 1964, pp. 1–26.

    MATH  MathSciNet  Google Scholar 

  11. Andreatta A.,Sui singrammi finiti tracciabili sul toro. Ist. Lomb. (Rend. Sc.) A 98, 1964, pp. 181–191.

    MATH  MathSciNet  Google Scholar 

  12. Andreatta A.,Problemi di tracciabilità per i singrammi doppi di altri. Ist. Lomb. (Rend. Sc.) A 103, 1969, pp. 50–58.

    MATH  MathSciNet  Google Scholar 

  13. Andreatta A.,Sui singrammi di permutazione tracciabili sul piano proiettivo. Ist. Lomb. (Rend. Sc.) A 103, 1969, pp. 412–417.

    MATH  MathSciNet  Google Scholar 

  14. Auslander L.—Brown T. A.—Youngs J. W. T.,The imbedding of graphs in manifolds. J. Math. Mech. 12, 1963, pp. 629–634.

    MathSciNet  Google Scholar 

  15. Balconi G.,Una caratterizzazione dei singrammi finiti con commutato sferico. Ist. Lomb. (Rend. Sc.) A 99, 1965, pp. 207–210.

    MATH  MathSciNet  Google Scholar 

  16. Battle J.—Harary F.—Kodama Y.—Young J. W. T.,Additivity of the genus of a graph. Bull. Amer. Math. Soc. 68, 1962, pp. 565–568.

    Article  MATH  MathSciNet  Google Scholar 

  17. Beineke L. W.—Harary F.,The genus of the n-cube. Canad. J. Math. 17, 1965, pp. 494–496.

    MATH  MathSciNet  Google Scholar 

  18. Coxeter H. S. M.,The Map-Coloring of Unorientable Surfaces. Duke Math. J. 10, 1943, pp. 293–304.

    Article  MATH  MathSciNet  Google Scholar 

  19. Dirac G. A.,Map Colour Theorems. Can. J. Math. 4, 1952, pp. 480–490.

    MATH  MathSciNet  Google Scholar 

  20. Dirac G. A.—Schuster S.,A Theorem of Kuratowski. Indag. Math. 16, 1954, pp. 343–348.

    MathSciNet  Google Scholar 

  21. Dirac G. A.,Map Colour Theorems Related to the Heawood Colour Formula. J. London Math. Soc. 31, 1956, pp. 460–471.

    Article  MATH  MathSciNet  Google Scholar 

  22. Dirac G. A.,On the Four-Color Conjecture. Proc. London Math. Soc. (3), 13, 1963, pp. 193–218.

    Article  MATH  MathSciNet  Google Scholar 

  23. Duke R.,The Genus, Regional Number, and Betti Number of a Graph. Canad. J. Math. 18, 1966, pp. 817–822.

    MATH  MathSciNet  Google Scholar 

  24. Edmonds J.,A Combinatorial Representation for Polyhedral Surfaces. Notices Amer. Math. Soc. 7, 1960, pp. 646.

    Google Scholar 

  25. Gross J. L.,The Genus of Nearly Complete Graphs-Case 6. I B M Watson Research Center Preprint New York, Yorktown Heights, 10 pp.

  26. Gross J. L.—Alpert S.,Branched Covering of Graph Imbeddings. I B M Res. Pamphlet R C 4216 New York, Yorktown heights, 18 pp.

  27. Gustin W.,Orientable Embedding of Cayley Graphs. Bull. Amer. Math. Soc. 69, 1963, pp. 272–275.

    Article  MATH  MathSciNet  Google Scholar 

  28. Harary F.—Karp R.—Tutte W. H.,A Criterion for Planarity of the Square of a Graph. Journal of Comb. Theory 2, 1967, pp. 395–405.

    Article  MATH  MathSciNet  Google Scholar 

  29. Heawood P. J.,Map Colour Theorem. Quart. J. Math. 24, 1890, pp. 332–338.

    Google Scholar 

  30. Kagno I. N.,A Note on the Heawood Color Formula. J. Math. Phys. 14, 1935, pp. 228–231.

    MATH  Google Scholar 

  31. Kagno I. N.,The Mapping of Graphs on Surfaces. J. Math. Phys. Massachusetts Inst. Technol. 16, 1937, pp. 46–75.

    MATH  Google Scholar 

  32. Kainen P. C.,Embeddings and Orientations of Graphs. Combinat. Struct. and their Appl. 193–196, New York, Gordon and Breach 1970.

    Google Scholar 

  33. Kotzig A.,Aus der Theorie der endlichen regularen Graphen dritten und vierten Grades. Casopis Pest. Mat. 82, 1957, pp. 76–92.

    MATH  MathSciNet  Google Scholar 

  34. Kuratowski C.,Sur le probleme des courbes gauches en topologie. Fund. Math. 15, 1930, pp. 238–271.

    Google Scholar 

  35. Landesman E. M.—Young J. W. T.,Smooth Solutions in Case I of the Heawood Conjecture for Non-orientable Surfaces. J. Comb. Theory 13, 1972, pp. 26–39.

    Article  MATH  Google Scholar 

  36. Lefschetz S.—Solomon,Planar Graphs and Related Topics. Proc. of the Nation. Ac. of Sc. 54, 1965, pp. 1763–1765.

    Article  MATH  MathSciNet  Google Scholar 

  37. Mac Lane S.—Saunders,A Structural Characterization of Planar Combinatorial Graphs. Duke Math. J. 3, 1937, pp. 460–472.

    Article  MATH  MathSciNet  Google Scholar 

  38. Nordhaus E.—Steward B.—White A.,On the Maximum Genus of a Graph. J. Comb. Th. II, 1971, pp. 258–267.

    Article  Google Scholar 

  39. Nordhaus E.—Steward B.—Ringeisen R. D.—White A.,A Kuratowski-Type Theorem for the Maximum Genus of a Graph. J. Comb. Th. B 12, 1972, pp. 260–267.

    Article  MATH  Google Scholar 

  40. Ringeisen R. D.,The Maximum Genus of a Graph. Ph. D. Thesis, Michigan State Univ. 1970.

  41. Ringeisen R. D.,Determining all Compact Orientable 2-Manifolds upon which Km,n has 2-cell Imbeddings. J. Comb. Th. 12, 1972, pp. 101–104.

    Article  MathSciNet  Google Scholar 

  42. Ringeisen R. D.,Upper and Lower Imbeddable Graphs. In «Graph Theory and Applications», Springer Verlag Berlin 1972.

    Google Scholar 

  43. Ringel G.,Das Geschlecht des Vollständigen Paaren Graphen. Abh. Math. Sem. Univ. Hamburg 28, 1965, pp. 139–150.

    Article  MATH  MathSciNet  Google Scholar 

  44. Ringel G.,Über Drei Kombinatorische Probleme am n-dimensionalen Würfel and Würfelgitter. Abh. Math. Sem. Univ. Hamburg 20, 1965, pp. 10–19.

    MathSciNet  Google Scholar 

  45. Ringel G.—Youngs J. W. T.,Solution of the Heawood Map-Coloring Problem. Proc. Nat. Acad. Sc. U.S.A., 60, 1968, pp. 438–445.

    Article  MATH  MathSciNet  Google Scholar 

  46. Ringel G.—Youngs J. W. T.,Solution of the Heawood Map-Coloring Problem Case 11. J. Comb. Th. 7, 1969, pp. 71–93.

    Article  MathSciNet  Google Scholar 

  47. Ringel G.—Youngs J. W. T.,Solution of the Heawood Map-Coloring Problem Case 8. J. Comb. Th. 7, 1969, pp. 353–363.

    Article  MathSciNet  Google Scholar 

  48. Ringel G.—Youngs J. W. T.,Solution of the Heawood Map-Coloring Problem Case 2. J. Comb. Th. 7, 1969, pp. 342–352.

    Article  MathSciNet  Google Scholar 

  49. Scotti F.—Torre A.,Grafi triangolati tracciabili sul toro. Ist. Lomb. (Rend. Sc.) A 110, 1976, pp. 520–530.

    MathSciNet  Google Scholar 

  50. Tazzi Cantalupi G.,Sui singrammi cubici hamiltoniani di corrispondenza inversa. Ist. Lomb. (Rend. Sc.) A 98, 1964, 319–326.

    MathSciNet  Google Scholar 

  51. Tazzi Cantalupi G.,Sul rango di un singramma cubico hamiltoniano di corrispondenza diretta. Ist. Lomb. (Rend. Sc.) A 99, 1965, pp. 895–903.

    MATH  MathSciNet  Google Scholar 

  52. Tazzi C. G.—Ricetti Z. B.,Singrammi triangolati sferici. Ist. Lomb. (Rend. Sc.) A 106, 1972, pp. 697–703.

    MATH  Google Scholar 

  53. Tazzi C. G.,Singrammi triangolati tracciabili sul piano proiettivo. Ist. Lomb. (Rend. Sc.) A 107, 1973, pp. 543–549.

    MATH  Google Scholar 

  54. Terry C. M.—Welch L. R.—Youngs J. W. T.,Solution of the Heawood Map-Coloring Problem Case 4. J. Comb. Th. 8, 1970, pp. 170–174.

    Article  MATH  MathSciNet  Google Scholar 

  55. Terry C. M.—Welch L. R.—Youngs J. W. T.,The Genus of K12s. J. Comb. Th. 2, 1967, pp. 43–60.

    Article  MATH  MathSciNet  Google Scholar 

  56. Tutte W. T.,On the Imbedding of Linear Graphs in Surfaces. Proc. London Math. Soc. (2) 51, 1949, pp. 474–483.

    MATH  MathSciNet  Google Scholar 

  57. Tutte W. T.,A Theorem on Planar Graphs. Trans. Amer. Math. Soc. 82, 1956, pp. 99–116.

    Article  MATH  MathSciNet  Google Scholar 

  58. Vazonyi E.,Graphen auf Flachen. Mat. Fiz. Lapok 44, 1937, pp. 133–163.

    Google Scholar 

  59. Ventre A. G. S.,Sull'immergibilità inferiore e superiore del grafo di Grötzsch. Riv. Mat. Univ. Parma (4) 1, 1975, pp. 227–231.

    MathSciNet  Google Scholar 

  60. Ventre A. G. S.,Sulla superiore immergibilità di grafi costituiti da blocchi superiormente immergibili. Boll. U.M.I. (4) 12, Suppl. Fasc. 3, 1975, pp. 388–397.

    MathSciNet  Google Scholar 

  61. Ventre A. G. S.,On the Genus and the Maximum Gnus of Graphs in Connection with the ones of their Blocks. Boll. U.M.I. (5) 14 A, 1977, pp. 626–633.

    MathSciNet  Google Scholar 

  62. Veress P.,Über nicht-ebene Graphen. Mat. Fiz. Lapok 44, 1940, pp. 34–47.

    MathSciNet  Google Scholar 

  63. Wagner K.,Zwei Bemerkungen über Komplexe. Math. Ann. 112, 1936, pp. 316–321.

    Article  MathSciNet  Google Scholar 

  64. Wagner K.,Über eine Erweiterung eines Satzes von Kuratowski. Deutsche Math., 2, 1937, pp. 280–285.

    Google Scholar 

  65. White A. T.,The Genus of the Complete Tripartite Graph Kmn,n,n. J. Comb. Th. 7, 1969, pp. 283–285.

    Article  MATH  Google Scholar 

  66. White A. T.,The Genus of Repeated Cartesian Products of Bipartite Graphs. Trans. Amer. Math. Soc. 151, 1970, pp. 393–404.

    Article  MATH  MathSciNet  Google Scholar 

  67. White A. T.,The Genus of the Cartesian Product of two Graphs. J. Comb. Th. 11, 1971, pp. 89–94.

    Article  Google Scholar 

  68. White A. T.,On the Genus of the Composition of two Graphs. Pac. J. Math. 41, 1972, pp. 275–279.

    MATH  Google Scholar 

  69. Whitney H.,Non Separable and Planar Graph. Trans. Amer. Math. Soc. 34, 1932, pp. 339–362.

    Article  MathSciNet  Google Scholar 

  70. Whitney H.—Hassler,Planar Graphs. Fundam. Mat. 21, 1933, pp. 73–84.

    MATH  Google Scholar 

  71. Youngs J. W. T.,Minimal Imbeddings and the Genus of a Graph. J. Math. Mech 12, 1963, pp. 303–316.

    MATH  MathSciNet  Google Scholar 

  72. Youns J. W. T.,Remarks on the Heawood Conjecture (Nonorientable Case). Bull. Am. Math. Soc. 74, 1968, pp. 347–353.

    Article  Google Scholar 

  73. Youngs J. W. T.,The Nonorientable Genus of Kn. Bull. Am. Math. Soc. 74, 1968, pp. 354–358.

    Article  MATH  MathSciNet  Google Scholar 

  74. Youngs J. W. T.,Solution of the Heawood Map-Coloring Problem Cases 3, 5, 6 and 9. J. Comb. Th. 8, 1970, pp. 175–219.

    Article  MATH  MathSciNet  Google Scholar 

  75. Youngs J. W. T.,The Heawood Map-Coloring Cases 1, 7 and 10. J. Comb. Th. 8, 1970, pp. 220–231.

    Article  MATH  MathSciNet  Google Scholar 

  76. Zaks J.,The Maximum Genus of Cartesian Products of Graphs. Canad. J. Math. 26, 1974, 1025–1035.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. del Politecnico di Milano, Milano, Italia

    Gabriella Tazzi Cantalupi

Authors
  1. Gabriella Tazzi Cantalupi
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

(Conferenza tenuta il 30 maggio 1978)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cantalupi, G.T. Sulla tracciabilita' di grafi finiti su superficie compatte. Seminario Mat. e. Fis. di Milano 48, 87–103 (1978). https://doi.org/10.1007/BF02925564

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02925564

Search

Navigation