Regular Homotopy Theory and the Construction of the Derived Graphs

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.
∗
jcarlos.kiihl@gmail.com
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