Metacirculant tournaments whose order is a product of two distinct primes

In this paper, we prove that non-circulant vertex-transitive tournaments of order pq, where p and q are distinct odd primes, are metacirculant tournaments (defined in Definition 2.1) satisfying some special conditions; see Theorem 1.2. So, in combination with the work in Jing Xu (2010) [11], a complete classification of vertex-transitive pq-tournaments is obtained. As a by-product, we construct examples of non-Cayley vertex-transitive pq-tournaments where q2|(p−1) in Example 2.5. Moreover, applying the classification of vertex-transitive pq-tournaments, we determine all 2-closed (in Wielandt's sense) odd-order transitive permutation groups of degree pq and show that each of them is the full automorphism group of some tournament.