Tetravalent one-regular graphs of order 2pq

Abstract

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this article a complete classification of tetravalent one-regular graphs of order twice a product of two primes is given. It follows from this classification that with the exception of four graphs of orders 12 and 30, all such graphs are Cayley graphs on Abelian, dihedral, or generalized dihedral groups.

An Algorithmic-type Classification of Tetravalent One-regular Graphs Using Computer Algebra Tools

“…Also, by [6,35,36,44,50, 51] every tetravalent one-regular graph of order pq or p 2 is a circulant graph. Furthermore, the classification of tetravalent one-regular graphs of order 3p 2 , 4p 2 , 5p 2 , 6p 2 and 2pq are given in [8,18,19,20,54]. Along this line we obtain Theorem 3.1, which is the main result of the paper.…”

4-valent graphs of order 6p^2 admitting a group of automorphisms acting regularly on arcs

“…We recall that such graphs are already classified when their orders are a prime or the product of two (not necessarily distinct) primes [3,21,22,26,29,28]. Moreover, for p and q primes, the classification of the 4-valent one-regular graphs of order 4p 2 or 2pq is given in [4,31]. In this context we prove the following.…”

Pentavalent symmetric graphs of order 2p 3

“…The classification of tetravalent s-transitive Cayley graphs on abelian groups were given in [38]. Zhou and Feng [42] gave a classification of tetravalent 1-regular graphs of order 2pq, and they also classified tetravalent s-transitive graphs of order 4p or 2p 2 in [39,40]. For the pentavalent symmetric graphs, Li and Feng [23] classified pentavalent 1-regular graphs of square free order, and Hua and Feng [17,18] classified pentavalent symmetric graphs of order 2pq or 8p.…”