Hamiltonian in graph theory

Question

Show that if G is bipartite with bipartition (x,y) where |x| is not equal to |y|, then G is non-hamiltonian.

Case 1. G is not connected (trivial) Case 2. G is connected

Is it ok to start Letting C be a Hamiltonian cycle of G?

Answer 1

Only if you word it correctly: you can assume for contradiction that there is a Hamilton cycle, then comment on its structure.

You can, however, comment on the structure of any cycle directly, without assuming Hamiltonicity -- for instance, if the cycle $(v_1,v_2,\ldots,v_m,v_1)$ is in $G$, how must the $v_i$ be related because of the bipartition condition? You can use this to come up with a bound for the longest cycle in $G$, and it will certainly be less than $n$.