Regular Homotopy Theory and the Construction of the Derived Graphs J. Carlos S. Kiihl∗ Abstract: In this paper we present an overview of the Regular Homotopy Theory for Digraphs, a survey of its main applications and the most recent developments. We show that the o-regular maps introduced by D. C. Demaria coincides with the pre-continuos maps if, in a natural way, we introduce in the class of the digraphs a structure of pre-topological space. We state and present the basic concepts and the fundamental results of this Homotopy Theory. New homotopical concepts and invariants associated to digraphs are stablished and, using these new tools, a new approach to the study of digraphs can be used, specially for the case of tournaments. For the Hamiltonian tournaments we have the concepts of: minimal cycles, cyclic characteristic, neutral and non-neutral vertices. One of the main purpose here is to stablish the properties of these concepts which are important in order to obtain structural characterizations for certain families of tournaments, when they are approached from a homotopical point of view. As some important applications we list some already known results about simply disconnected, normal, Douglas tournaments and Moon tournaments. Then we give a complete study of the Hamiltonian 5-tournaments and 6-tournaments (having the maximal number of non-neutral vertices) in terms of minimal 3-cycles. This has led to the discovery of the construction of the Derived Graph associated to a Hamiltonian tournament. This most recent construction is offering us a powerfull tool to continue our studies on the structural caracterization of certain families of tournaments. Keywords: Regular Homotopy Theory, Digraphs, Tournaments, Cycles, Derived Graph. MSC (2010) - 55Q99,05C20. ∗ [email protected] 1 References [1] BEINEKE, L. W. and REID, K. B., Tournaments–Selected Topics in Graph Theory, Edited by L. W. Beineke and R. J. Wilson, Academic Press, New York (1978), 169–204. [2] BURZIO M. and DEMARIA D.C., A normalization theorem for regular homotopy of finite directed graphs, Rend. Circ. Matem. Palermo, (2), 30 (1981), 255–286. [3] BURZIO M. and DEMARIA D.C., The first normalization theorem for regular homotopy of finite directed graphs, Rend. Ist. Mat. Univ. Trieste, 13 (1981), 38–50. [4] BURZIO M. and DEMARIA D.C., The second and third normalization theorem for regular homotopy of finite directed graphs, Rend. Ist. Mat. Univ. Trieste, 15 (1983), 61–82. [5] BURZIO, M. and DEMARIA, D. C., Duality theorem for regular homotopy of finite directed graphs, Rend. Circ. Mat. Palermo, (2), 31 (1982), 371–400. [6] BURZIO, M. and DEMARIA, D. C., Homotopy of polyhedra and regular homotopy of finite directed graphs, Atti IIo Conv. Topologia, Suppl. Rend. Circ. Mat. Palermo, (2), no¯ 12 (1986), 189–204. [7] BURZIO, M. and DEMARIA, D. C., Characterization of tournaments by coned 3–cycles, Acta Univ. Carol., Math. Phys., 28 (1987), 25–30. [8] BURZIO, M. and DEMARIA, D. C., On simply disconnected tournaments, Proc. Catania Confer. Ars Combinatoria, 24 A (1988), 149–161. [9] BURZIO, M. and DEMARIA, D. C., On a classification of hamiltonian tournaments, Acta Univ. Carol., Math. Phys., 29 (1988), 3–14. [10] BURZIO, M. and DEMARIA, D. C., Hamiltonian tournaments with the least number of 3-cycles, J. Graph Theory 14 (6) (1990), 663–672. [11] CAMION P., Quelques porpri´et´e des chemins et circuits Hamiltoniens ´ dans la th´eorie des graphes, Cahiers Centre Etudes Rech. Oper., vol 2 (1960), 5–36. 2 ˘ [12] CECH E.,Topological Spaces, Interscience, London (1966). [13] DEMARIA D. C.; GARBACCIO BOGIN R., Homotopy and homology in pretopological spaces, Proc. 11th Winter School, Suppl. Rend. Circ. Mat. Palermo, (2), No¯ 3 (1984), 119-126. [14] DEMARIA D.C.; GANDINI, P. M., Su una generalizzazione della teoria dell’omotopia, Rend. Sem. Mat. Univ. Polit. Torino , 34 (1975 - 76). [15] DEMARIA D.C.; GIANELLA G.M., On normal tournaments , Conf. Semin. Matem. Univ. Bari ,vol. 232 (1989), 1-29. [16] DEMARIA D.C.; GUIDO C., On the reconstruction of normal tournaments, J. Comb. Inf. and Sys. Sci., vol. 15 (1990), 301-323. [17] DEMARIA D.C.; KIIHL J.C. S., On the complete digraphs which are simply disconnected, Publicacions Mathem`atiques, vol. 35 (1991), 517525. [18] DEMARIA D.C.; KIIHL J.C. S., On the simple quotients of tournaments that admit exactly one hamiltonian cycle, Atti Accad. Scienze Torino, vol. 124 (1990), 94-108. [19] DEMARIA D.C.; KIIHL J.C. S., Some remarks on the enumeration of Douglas tournaments, Atti Accad. Scienze Torino , vol. 124 (1990), 169-185. [20] DOUGLAS R.J., Tournaments that admit exactly one Hamiltonian circuit, Proc. London Math. Soc., 21, (1970), 716-730. [21] GANDINI, P. M., Sull’omotopia per pseudoarchi, Rend. Sem. Mat. Univ. Polit. Torino , 33 (1974 - 75). [22] GIANELLA, G. M., Sull’omotopia per quasiarchi, Rend. Sem. Mat. Univ. Polit. Torino , 31 (1971 - 72, 1972 - 73). [23] GUIDO C., Structure and reconstruction of Moon tournaments, J. Comb. Inf. and Sys. Sci., vol. 19 (1994), 47-61. [24] GUIDO C., A larger class of reconstructable tournaments, Dicrete Math. 152 (1996), 171-184. 3 [25] GUIDO C.; KIIHL J.C.S.; OLIVEIRA, J. P. M.; BORRI, M., Some remarks on non-reconstructable tournaments, (to appear). [26] KIIHL, J. Carlos S.;GONC ¸ ALVES, A. C., On Digraphs and their Quotients, Revista Iluminart, Volume 9 (2012), 195 - 208. [27] KIIHL, J. Carlos S.; GUADALUPE, Irwen Valle, Either Digraphs or Pre-Topological Spaces?, Revista Iluminart, Volume 6 (2011), 129 - 147 [28] KIIHL, J. Carlos S.; TIRONI, Gino; GONC ¸ ALVES, A. C., The Minimal Cycles, Neutral and Non-Neutral Vertices in Tournaments, Revista Iluminart, Volume 10 (2013), 213 - 238. [29] KIIHL, J. Carlos S.; LIMA, F. M. B.; OLIVEIRA, J. P. M.; GONC ¸ ALVES, A. C., 6-Tournaments having a minimal cycle of length four, Revista Iluminart, Volume 12 (2013), 179 - 192. [30] KIIHL, J. Carlos S.; GONC ¸ ALVES, Alexandre C., Hamiltonian Tournaments and Associated 3-Cycles Graphs, (to appear). [31] KIIHL, J. Carlos S.;TIRONI, Gino, Non-Coned Cycles: A New Approach to Tournaments, Revista Iluminart, Volume 7 (2011), 98 - 109. [32] MOON J.W., Topics on Tournaments, Holt, Rinehart and Winston, New York (1978). [33] MOON J.W., On subtournaments of a tournament, Canad. Math. Bull., vol. 9 (3) (1966), 297-301. [34] MOON J.W., Tournaments whose subtournaments are irreducible or transitive, Canad. Math. Bull., vol. 21 (1) (1979), 75-79. ¨ ˘ RIL ˘ J.; PELANT J., Either tournaments or al[35] MULLER, V.; NESET gebras?, Discrete Math., 11 (1975), 37–66. [36] STOCKMEYER P.K., The reconstruction conjecture for tournaments, in “Proceedings, Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing” (F.Hoffman et al., Eds.), 561-566, Utilitas Mathematica, Winnipeg, (1975). [37] STOCKMEYER P.K., The falsity of the reconstruction conjecture for tournaments, J. Graph Theory 1 (1977), 19-25. 4
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