yes, here exist atleast a cycle with odd no.of vertices and there exist atleast a cycle with even no.of vertices.

name them as A,B,C,D,E,F,G,H,I as row by row

A-B-C-F-E-H-G-D-A ------> cycle with EVEN no.of vertices

A-B-C-F-I-A -----> cycle with ODD no.of vertices

name them as A,B,C,D,E,F,G,H,I as row by row

A-B-C-F-E-H-G-D-A ------> cycle with EVEN no.of vertices

A-B-C-F-I-A -----> cycle with ODD no.of vertices