2-arc-transitive regular covers of $K_{n,n}$ having the covering transformation group $\mathbb{Z}_p^2$
type: Item
instance of: scholarly article
title: 2-arc-transitive regular covers of $K_{n,n}$ having the covering transformation group $\mathbb{Z}_p^2$
zbMATH Open document ID: 1424.05284
author: Wenqin Xu
author: Shaofei Du
author: Guiying Yan
published in: Combinatorica
publication date: 2018-09-24T00:00:00Z
review text: As a part of an ongoing project of classifying $2$-arc-transitive graphs, the authors choose to focus on the classification of $2$-arc-transitive lifts of highly symmetric graphs by elementary abelian groups. Specifically, the base graphs of their choice are the complete bipartite graphs $K_{n,n}$, and the covering transformation groups are the groups $Z_{p}^2$. Both choices reflect the current state of knowledge since the regular covers of complete graphs $K_n$ by covering tranformation groups $Z_p^2$ as well as the covers of $K_{n,n}$ by cyclic covering transformation groups have already been classified. \par They succeed at classifying all regular covers of $K_{n,n}$ whose covering transformation group is $Z_p^2$ and whose fibre-preserving automorphism group acts $2$-arc-transitively. They show that these covers are of three types all of which they define via a voltage assignment. The first family is obtained via voltage assignments of $K_{p,p}$ with voltages selected from $Z_p^2$ ($p \geq 5$ a prime), the second by voltages assigned to $K_{p^s,p^s}$ using dot-products of vectors from an $s$-dimensional vector space over $F_p$ ($s \geq 2$ and $p$ a prime), and the third by voltages from $Z_p^2$ assigned to $K_{3,3}$ ($p$ a prime). Only the third family has been previously known.\par The proof is quite involved, over 14 pages long, and essentially group-theoretical.
Mathematics Subject Classification ID: 05E15
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 6941937
zbMATH Keywords: $2$-arc-transitive graphs
zbMATH Keywords: automorphism groups
zbMATH Keywords: regular covers
reviewed by: Robert Jajcay
MaRDI profile type: Publication
full work available at URL: s00493-016-3511-x
OpenAlex ID: W2602705146
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DOI: 10.1007/S00493-016-3511-X
description: scientific article; zbMATH DE number 6941937