Tetravalent half-arc-transitive graphs of order 2pq

A graph is half-arc-transitive if its automorphism group acts transitively on vertices and edges, but not on arcs. It is known that for a prime p there is no tetravalent half-arc-transitive graphs of order p or p^2. Xu [M.Y. Xu, Half-transitive graphs of prime-cube order, J. Algebraic Combin. 1 (1992) 275-282] classified the tetravalent half-arc-transitive graphs of order p^3. As a continuation, we classify in this paper the tetravalent half-arc-transitive graphs of order p^4. It shows that there are exactly p-1 nonisomorphic connected tetravalent half-arc-transitive graphs of order p^4 for each odd prime p.

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