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