In connection with the classification problem for non-Cayley tetravalent metacirculant graphs, three families of special tetravalent metacirculant graphs, denoted by Φ1, Φ2 and Φ3, have been defined . It has also been shown  that any non-Cayley tetravalent metacirculant graph is isomorphic to a union of disjoint copies of a non-Cayley graph in one of the families Φ1, Φ2 or Φ3. A natural question raised from the result is whether all graphs in these families are non-Cayley. We have proved recently in  that every graph in Φ2 is non-Cayley. In this paper, we show that every graph in Φ1 is also a connected non-Cayley graph and find an infinite class of connected non-Cayley graphs in the family Φ3.
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