Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Consistent cycles in 1 -arc-transitive graphs 2. | Consistent cycles in 1-arc-transitive graphs Marko Boben Faculty of Computer and Information Science University of Ljubljana and Institute of Mathematics Physics and Mechanics Trzaska 25 SI-1000 Ljubljana Slovenia Stefko Miklavic University of Primorska Institute for Natural Science and Technology Mestni trg 2 SI-6000 Koper Slovenia Primoz Potocnik Faculty of Mathematics and Physics University of Ljubljana and Institute of Mathematics Physics and Mechanics Jadranska 19 SI-1000 Ljubljana Slovenia Submitted Nov 7 2008 Accepted Dec 15 2008 Published Jan 7 2009 Mathematics Subject Classification 05C25 05C38 Abstract A directed cycle C of a graph is called 1 -consistent if there exists an automorphism of the graph which acts as a k-step rotation of C. These cycles have previously been considered by several authors in the context of arc-transitive graphs. In this paper we extend these results to the case of graphs which are vertex-transitive edge-transitive but not arc-transitive. 1 Introduction A long neglected result of J. H. Conway 1 2 states that an arc-transitive graph of valence d has exactly d 1 orbits of the so-called consistent directed cycles where directed cycle is consistent whenever there exists an automorphism of a graph which acts on the cycle as a one step rotation. The original result of Conway first appeared in 1 and several THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R5 1 rigorous proofs together with some generalizations of the result were recently provided in 5 9 10 . In this paper we extend the result of Conway to graphs r admitting a group of automorphisms G acting transitively on the vertices and the edges but intransitively on the directed edges of r. Moreover we consider the so called k -consistent cycles of r that is cycles which may not allow a one-step rotation but rather a k-step rotation for k 2. If such a graph r has girth at least 2k 1 then we show that the
