Graphs and Combinatorics

, Volume 18, Issue 4, pp 795-802

First online:

On Non-Cayley Tetravalent Metacirculant Graphs

  • Ngo Dac TanAffiliated withHanoi Institute of Mathematics, P.O. Box 631 Bo Ho, 10 000 Hanoi, Vietnam e-mail:

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 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.