When do graphs have a Hamiltonian Cycle and H-Path?

what are the formulas for Hamiltonian Cycle and Hamiltonian Path existence within a graph? What are the formulas for when complete graphs, complete bipartite etc have H-cycles?

A graph has a H-cycle IFF:

1. has at least one cycle,
2. must have even # of vertices?

Are these correct?

Complete graphs:

H path ( K_n ): $n \geq 2$

H cycle ( K_n ): $n \geq 2$

Complete Bipartite graphs:

H path ( K_m,n ): $|m-n| \leq 1$

H cycle ( K_m,n ): $ m = n > 1$