On Non-Cayley Tetravalent Metacirculant Graphs

Abstract.

 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 [11]. It has also been shown [11] 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 [12] 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.