On cubic non‐Cayley vertex‐transitive graphs

Abstract

Abstract In 1983, the second author [D. Marušič, Ars Combinatoria 16B (1983), 297–302] asked for which positive integers n there exists a non‐Cayley vertex‐transitive graph on n vertices. (The term non‐Cayley numbers has later been given to such integers.) Motivated by this problem, Feng [Discrete Math 248 (2002), 265–269] asked to determine the smallest valency ϑ( n ) among valencies of non‐Cayley vertex‐transitive graphs of order n . As cycles are clearly Cayley graphs, ϑ( n )⩾3 for any non‐Cayley number n . In this paper a goal is set to determine those non‐Cayley numbers n for which ϑ( n ) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non‐Cayley vertex‐transitive graphs of order n . It is known that for a prime p every vertex‐transitive graph of order p , p 2 or p 3 is a Cayley graph, and that, with the exception of the Coxeter graph, every cubic non‐Cayley vertex‐transitive graph of order 2 p , 4 p or 2 p 2 is a generalized Petersen graph. In this paper the next natural step is taken by proving that every cubic non‐Cayley vertex‐transitive graph of order 4 p 2 , p >7 a prime, is a generalized Petersen graph. In addition, cubic non‐Cayley vertex‐transitive graphs of order 2 p k , where p >7 is a prime and k ⩽ p , are characterized. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 77–95, 2012