ar X iv : m at h / 06 06 58 5 v 1 [ m at h . C O ] 2 3 Ju n 20 06 HAMILTONICITY OF VERTEX - TRANSITIVE GRAPHS OF ORDER 4 p

It is shown that every connected vertex-transitive graph of order 4p, where p is a prime, is hamiltonian with the exception of the Coxeter graph which is known to possess a Hamilton path. In 1969, Lovász [22] asked if every finite, connected vertex-transitive graph has a Hamilton path, that is, a path going through all vertices of the graph. With the exception of K 2 , only four connected vertex-transitive graphs that do not have a Hamilton cycle are known to exist. These four graphs are the Petersen graph, the Coxeter graph and the two graphs obtained from them by replacing each vertex by a triangle. The fact that none of these four graphs is a Cayley graph has led to a folklore conjecture that every Cayley graph is hamiltonian for the current status of this conjecture). Coming back to vertex-transitive graphs, it was shown in [14] that, with the exception of the Petersen graph, a connected vertex-transitive graph whose automorphism group contains a transitive subgroup with a cyclic commutator subgroup of prime-power order, is hamiltonian. Furthermore, for connected vertex-transitive graphs of orders p, 2p (except for the Petersen graph), 3p, p 2 , p 3 , p 4 and 2p 2 it was shown that they are hamiltonian (Throughout this paper p will always denote a prime number.) On the other hand, connected vertex-transitive graphs of orders 4p and 5p are only known to have Hamilton paths (see [27, 28]). It is the object of this paper to complete the analysis of hamiltonian properties of vertex-transitive graphs of order 4p by proving the following result. Theorem 1.1 With the exception of the Coxeter graph, every vertex-transitive graph of order 4p, where p is a prime, is hamiltonian.