Graphs corresponding to closed walks of length k are known as k-cyclic graphs, or C_k-graphs for short. C_k-graphs are connected by definition. The numbers of C_k-graphs for k = 3, 4, ... are 1, 3, 3, 10, 12, 35, 58, 160, 341, 958, 2444, 7242, 21190, 67217, 217335, ... (OEIS A081809; FlowProblems), the first few of which are illustrated above. It appears that every connected simple graph on more than one node is C_k for some value of k. For example, every connected graph on six or fewer nodes with the exception of the complete graph K_6 is C_k for some 3<=k<=17.

circuit | cyclic graph | graph cycle | Hamiltonian walk | trail | walk