For any $z \times y$ grid graph with $z$ rows consisting of $y$ vertices, find the path with the greatest length in such a graph. A path is considered where no vertices and no edges are repeated. Give some explanation as to why this is the longest path.
Let $z, y \ge 2$
What I have so far:
If we have a grid of $z \times y$ then there are $2zy - z -y$ edges in the grid total. I think it's safe to assume that there are no paths containing all edges. Now how can I determine the upper bound of the longest path and then prove there exists some path having this length?