Discrete Mathematics 231 (2001) 337–349
www.elsevier.com/locate/disc
Construction of k-arc transitive digraphs
SÂonia P. Mansilla
∗;1
, Oriol Serra
Department of Applied Mathematics and Computer Science, Universitat Polit
Â
ecnica de Catalunya,
C. Gran Capit
Â
a, s/n-M
Â
odul C3, Campus Nord, E-08034 Barcelona, Spain
Received 14 July 1999; revised 30 May 2000; accepted 7 August 2000
Abstract
A digraph is k-arc transitive ifit has a group ofautomorphisms which acts transitively on the
set of k-arcs. Unlike the undirected case, in which the cycles are the only k-arc transitive ÿnite
graphs for k¿8, there are k-arc transitive ÿnite digraphs with arbitrary out-degree for every
positive integer k. We show that every regular ÿnite digraph admits a covering digraph which
is k-arc transitive. The result provides a technique to construct k-arc transitive digraphs from
arbitrary regular digraphs. Some examples are given from complete graphs with and without
loops.
c
2001 Elsevier Science B.V. All rights reserved.
Keywords: Cayley digraph; k-arc transitive digraph; Covering digraph
1. Introduction
For a positive integer k,ak-arc ofa digraph =(V; E) is a sequence (x
0
;:::;x
k
)of
k + 1 vertices of such that, for each 06i¡k; (x
i
;x
i+1
) is an arc ofthe digraph. A
digraph is k-arc transitive ifit has an automorphism group G¡Aut which acts
transitively on k-arcs.
The corresponding notion for undirected graphs led to remarkable results. A well-
known result by Tutte [19] states that ÿnite cubic graphs cannot be k-arc transitive
for k¿5. Weiss [20] proved several years later that the only ÿnite connected k-arc
transitive graphs with k¿8 are the cycles.
The situation is dierent in the directed case. Praeger [14] gave inÿnite families
of k-arc transitive digraphs for each positive integer k and each out-degree v; and
new constructions were given by Conder et al. [4]. Recently, Cameron et al. [3] gave
Work supported by the Spanish Research Council CICYT under project TIC-97-0963, and the Catalan
Research Council under grant 1998 SGR00119.
∗
Corresponding author.
E-mail addresses: sonia@mat.upc.es (S.P. Mansilla); oriol@mat.upc.es (O. Serra).
1
Partially supported by the Comissionat per a Universitats i Recerca ofthe Generalitat de Catalunya under
Grant 1997FI-693.
0012-365X/01/$ - see front matter
c
2001 Elsevier Science B.V. All rights reserved.
PII: S0012-365X(00)00331-9