Abstract: Rm-edge cut is such an edge cut that separates a connected graph into a disconnected one with no component having order less than m. Let G be a connected graph with order at least 2m. Write c(G) for the circumference of graph G, i.e., the length of the longest cycle in G. It is proved in this paper that graph G contains Rm-edge cuts if c(G) ≥m+1. The lower bound of c(G) is sharp to some extent.