(*  Hopefully a critically hamiltonian graph
It is the Cayley graph of the additive group Z_53
with generating set {+-1, +-20}
*)

k=53

ponot = Range[k] ;


Crit1=MakeGraph[ ponot,
  (
    (
      (#2==Mod[#1+1, k])
    )||
    (
      (#2==Mod[#1+k-1, k])
    )||
    (
      (#2==Mod[#1+20, k])
    )||
    (
      (#2==Mod[#1+k-20, k])
    )
  )&, Type -> Undirected     ];