Bipartite graph problem

Question

A mouse wants to eat a 3*3*3 cube of cheese, in which there is a cherry in the exact center of the cube. It begins at a corner and, at each step, eats a whole 1*1*1 cube, before going on to an adjacent 1*1*1 cube (i.e., which shares a face with it). Can the mouse eat the 1*1*1 cube containing the cherry last (for dessert), i.e., can it end in the center of the 3*3*3 cube? I want an answer to this problem modelling it as a bipartite graph.Any suggestions ??

Answer 1

Yes, I suggest not to tease a pure hungry mouse, whether can it eat through a hamiltonian path on a bipartite graph with parts of $13$ and $14$ vertices, starting in a bigger part and ending in the smaller.