Reachability relations in digraphs

Primož Šparl
University of Ljubljana and University of Primorska, Slovenia

Aleksander Malnič
University of Ljubljana and University of Primorska, Slovenia

Primož Potočnik
University of Ljubljana and University of Primorska, Slovenia

Norbert Seifter
Montanuniversitaet Leoben, Austria

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Minisymposium: STRUCTURE AND PROPERTIES OF VERTEX-TRANSITIVE GRAPHS

Content: In 2002 Maru\v si\v c and Poto\v cnik introduced an infinite family of relations (now called {\em reachability relations}) on (finite) digraphs and proved various interesting results regarding them. In 2008 these relations were studied in the context of infinite digraphs by Malni\v c, Maru\v si\v c, Seifter, \v Sparl and Zgrabli\'c, again producing some nice results. The interplay of various (di)graph theoretic properties and properties of these relations on infinite (vertex-transitive) digraphs was further investigated in a paper by Seifter and Trofimov from 2009. In this talk we present a selection of the results on reachability relations from the above mentioned papers, together with a recent result by Malni\v c, Poto\v cnik, Seifter and \v Sparl, linking the properties of reachability relations of a given vertex-transitive digraph admitting a nilpotent normal subgroup of automorphisms to the nilpotency class of this subgroup. We also present some preliminary results from a work in progress regarding the potential use of these relations in the polycirculant conjecture problem for infinite digraphs. This is joint work with Aleksander Malni\v c, Primo\v z Poto\v cnik and Norbert Seifter.

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