Nilpotent families of endomorphisms of (P(V)+,∪)

  • Authors:

  • D. Fon-Der-Flaass;A. Kostochka;J. Nesetril;A. Raspaud;E. Sopena

  • Affiliations:

  • Institute of Mathematics, Novosibirsk 630090, Russia;Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois/ and Institute of Mathematics, Novosibirsk 630090, Russia;Department of Applied Mathematics, Institute of Theoretical Computer Sciences (ITI), Charles University, Prague, Czech Republic;LaBRI, Universite Bordeaux I, 33405 Talence Cedex, France;LaBRI, Universite Bordeaux I, 33405 Talence Cedex, France

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2002

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Abstract

A directed graph G = (V,A) is k-nice if for every u, υ,∈ V (allowing t = υ), and for every orientation of the edges of an undirected path of length k, there exists a u - υ walk of length k in G whose orientation coincides with that of the given path. A graph is nice if it is k-nice for some k. We generalize this notion using the notion of a nilpotent semigroup of endomorphisms of (P(V)+, ∪), and consider two basic problems:(1) find bounds for the nilpotency class of such semigroups in terms of their generators (in the language of graphs: provided that a graph G on n vertices is nice, find the smallest k such that G is k-nice);(2) find a way to demonstrate non-nilpotency of such semigroups (find as simple as possible characterization of non-nice graphs).