Hamilton cycles in primitive graphs of order 2rs

Abstract

After long term efforts, it was recently proved by Du, Kutnar and Marušič in 2021 that except for the Petersen graph, every connected vertex-transitive graph of order rs has a Hamilton cycle, where r and s are primes. A natural topic is to solve the hamiltonian problem for connected vertex-transitive graphs of 2rs. This topic is quite nontrivial, as the problem is still unsolved even for that of r = 5. In this paper, it is shown that except for the Coxeter graph, every connected vertex-transitive graph of order 2rs contains a Hamilton cycle, provided the automorphism group acts primitively on vertices.