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... 

HINT: You simply need to explain carefully how to produce the desired paths.


*

*There is always a Hamilton path that simply traverses the rows in alternate directions.  

*If, say, $m$ is even, as in your example, you can generalize the following idea (which I’ve left slightly incomplete). The evenness of $m$ is what makes the idea work.
  *-->*-->*-->*-->*  
                  | 
                  V  
  *   *<--*<--*<--*  
      |  
      V  
  *   *-->*-->*-->*  
                  |  
                  V  
  *<--*<--*<--*<--*

