Hamilton paths/cycles in grid graphs Let G be a grid graph with m rows and n columns, i.e. m = 4, n = 7 is shown here: 

For what values of m and n does G have a Hamilton path, and for what values of m and n does G have a Hamilton cycle? 
So far I've figured out that a grid graph always has a Hamilton path, and has a Hamilton cycle when at least one of m or n is even. I'm struggling to provide justification as to why this is true... 

To prove why it works when $mn$ is even and $m, n > 1$ it should be enough to provide a construction of a Hamiltonian cycle as Brain did in his answer.
To prove why there is no such cycle when $mn$ is odd consider the following: color each vertex as "black" when the sum of its coordinates is even otherwise as "white". In this way you get a checkerboard pattern and no likewise colored vertexes are connected. Since the total number of vertexes is odd the start and endpoints of a Hamiltonian path must have the same colors, therefore they can not be connected.
