Abstract. Let c(G) denote the number of cycles in G and let Q be a class of graphs. By cn(Q) we denote the minimum number c(G), where G is a graph from Q with n vertices. We consider the following classes of graphs:

  1. Qk - the class of k-connected graphs,
  2. Qk,d - the class of k-connected graphs with minimum degree d,
  3. Qk,d,D - the class of k-connected graphs with minimum degree d and maximum degree D.
We give polynomial upper bounds on cn(Qk) and cn(Qk,d), and lower bounds on cn(Qk,d) and cn(Qk,d,D).