Sunto
Un grafo orientatoG viene considerato come un prespazio (spazio di chiusura di Čech) ed in tal modo si possono costruire i gruppi di omotopiaQ n (G) di G. Inoltre si prova che per un torneoT le seguenti tre condizioni sono equivalenti:
-
1)
il gruppo fondamentaleQ 1(T) è non banale;
-
2)
T è la composizione di un torneo altamente regolare;
-
3)
T contiene almeno un 3-ciclo non proiettato, mentre ogni suo 3-ciclo proiettato appartiene ad una componente.
Summary
The aim of this survey is to give some applications of homotopy theory to directed graphs. At first we define the homotopy groupsQ n (G) of a directed graphG, by consideringG as a prespace (Čech closure space). Then a class of hamiltonian tournaments is characterized by the following equivalent conditions for a tournamentT:
-
1)
the fundamental groupQ 1(T) ofT is not trivial;
-
2)
T is the composition of any tornaments with a nontrivial highly regular tournament;
-
3)
each coned 3-cycle ofT belongs to a component ofT; moreover inT there is at least one nonconed 3-cycle.
Bibliografia
Burzio M. andDemaria D. C.,A normalization theorem for regular homotopy of finite directed graphs. Rend. Circ. Mat. Palermo, (2), 30 (1981), 255–286.
Burzio M. andDemaria D. C.,The first normalization theorem for regular homotopy of finite directed graphs. Rend. Ist. Mat. Univ. Trieste, 13 (1981), 38–50.
Burzio M. andDemaria D. C.,Duality theorems for regular homotopy of finite directed graphs. Rend. Circ. Mat. Palermo, (2), 31 (1982), 371–400.
Burzio M. andDemaria D. C.,The second and third normalization theorems for regular homotopy of finite directed graphs. Rend. Ist. Mat. Univ. Trieste, 15 (1983), 61–82.
Burzio M. andDemaria D. C.,Homotopy of polyhedra and regular homotopy of digraphs. Atti II° Conv. Topologia, Suppl. Rend. Circ. Mat. Palermo, (2). n. 12 (1986), 189–204.
Burzio M. andDemaria D. C.,On simply disconnected tournaments. To appear in Ars Combin. (Waterloo, Ont.).
Burzio M. andDemaria D. C.,Characterization of tournaments by coned 3-cycles. To appear in Acta Univ. Carolin., Math. Phys. (Prague).
Čech E. Topological spaces. Interscience, London (1966).
Demaria D. C.,Teoremi di normalizzazione per l’omotopia regolare dei grafi. Rend. Semin. Mat. Fis. Milano, 46 (1976), 139–161.
Demaria D. C.,Relazioni tra l’omotopia regolare dei grafi e l’omotopia classica dei poliedri. Conf. Semin. Mat. Univ. Bari, 153 (1978), 1–30.
Demaria D. C. andGarbaccio Bogin R.,Homotopy and homology in pretopological spaces. Proc. 11th Winter School, Suppl. Rend. Circ. Mat. Palermo, (2), n. 3 (1984), 119–126.
Harary F.,Graph theory. Addison-Wesley, Reading (Mass.), 1969.
Müller V., Nešetřil J. andPelant J.,Either tornaments or algebras? Discrete Math, 11 (1975), 37–66.
Author information
Authors and Affiliations
Additional information
Classificazioni dell’AMS: 55Q99, 05C20.
Rights and permissions
About this article
Cite this article
Demaria, D.C. Su alcune applicazioni dell’omotopia ai grafi orientati. Seminario Mat. e. Fis. di Milano 57, 183–202 (1987). https://doi.org/10.1007/BF02925050
Issue Date:
DOI: https://doi.org/10.1007/BF02925050