Highly arc-transitive digraphs with no homomorphism onto \(\mathbb Z\)
type: Item
instance of: scholarly article
title: Highly arc-transitive digraphs with no homomorphism onto \(\mathbb Z\)
zbMATH Open document ID: 1012.05080
author: Aleksander Malnič
author: Dragan Marušič
author: Norbert Seifter
author: Boris Zgrablić
published in: Combinatorica
publication date: 2003-05-18T00:00:00Z
review text: The authors' work is based on three previous papers: \textit{P. J. Cameron, C. E. Praeger} and \textit{N. C. Wormald} [Combinatorica 13, 377-396 (1993; Zbl 0793.05065)], \textit{C. E. Praeger} [Australas. J. Comb. 3, 207-210 (1991; Zbl 0758.05055)] and \textit{D. M. Evans} [Eur. J. Comb. 18, 281-286 (1997; Zbl 0873.05050)]. An \(s\)-arc of \(D\) is a walk on \(s+1\) vertices. A walk \(W\) is alternating, if the directions of edges are alternating, while one travels through \(W\). The digraph \(D\) is \(s\)-arc transitive if its automorphism group is transitive on the set of \(s\)-arcs, while \(D\) is said to be highly arc transitive if it is \(s\)-arc transitive for all \(s >0\). A digraph has property \(Z\) if it has a homomorphism onto the two-way infinite path \(Z\). The main result of the paper is the exhibition of highly arc-transitive digraphs for all odd \(n \geq 3\) without property \(Z\) that also satisfy that all in and out degrees are 2, and the reachability equivalance classes induce alternating cycles of length \(2n\).
reviewed by: András Pluhár
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 05C20
zbMATH DE Number: 1912036
zbMATH Keywords: highly arc-transitve
zbMATH Keywords: alternating
zbMATH Keywords: homomorphism
MaRDI profile type: Publication
full work available at URL: s004930200022
OpenAlex ID: W2004980855
DOI: 10.1007/S004930200022
description: scientific article; zbMATH DE number 1912036