Recent trends and future directions in vertex-transitive graphs

Abstract

A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade.Keywords: Vertex-transitive graph, arc-transitive graph, half-arc-transitive graph, Hamilton cycle, Hamilton path, semiregular group, (im)primitive group.Math. Subj. Class.: 05C25, 20B25 1 In the beginning Vertex-transitive graphs, that is, graphs whose automorphisms groups acts transitively on the corresponding vertex sets, have been an active topic of research for a long time now. Much of this interest is due to their suitability to model scientific phenomena when symmetry is an issue. It is the aim of this article to discuss recent developments surrounding two well known open problems in vertex-transitive graphs -the Hamilton path/cycle problem and the semiregularity problem -as well as the general question addressing structural properties of arc-transitive and half-arc-transitive graphs. In doing so we will also try to contemplate possible directions this area of research is likely to take in the near future. * Supported by "Agencija za raziskovalno dejavnost Republike Slovenije", research program P1-0285.E-mail addresses: klavdija.kutnar@upr.si (Klavdija Kutnar), dragan.marusic@upr.si (Dragan Marušič) Copyright c 2008 DMFA -založništvo K. Kutnar, D. Marušič: Recent Trends and Future Directions in Vertex-Transitive Graphs 113The article is organized as follows. In Section 2 we discuss the problem, posed by the second author in 1981 (see [79]) who asked if it is true that a vertex-transitive digraph contains a nonidentity automorphism with all orbits of equal length, in short, a semiregular automorphism. Section 3 gives a quick overview of the problem, posed by Lovàsz in 1969 (see [72]), who asked if it is true that every connected vertex-transitive graph contains a Hamilton path, thus motivating a great deal of research into vertex-transitive graphs in the following decades. In Section 4, imprimitivity, one of the most fundamental concepts in the theory of permutation groups, is considered with special emphasis given to a recently developed theory which links the existence of blocks of imprimitivity in vertex-transitive graphs, having an abelian semiregular subgroup of automorphisms, to certain conditions that need to be satisfied in the corresponding quotient graph relative to the orbits of such a subgroup. Finally, in Section 5 we deal with some recent structural results on arc-transitive and half-arc-transitive graphs, as well as their link to some open problems in the theory of configurations.For group-theoretic terms not defined here we refer the reader to [124]. Semiregularit...

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