u/frogkabobs

The graph Gₙ consists of vertices (x,y) for integers 1≤x≤n and 1≤y≤3 and edges between (x,y) and (x',y') iff x=x' or both |x-x'|=1 and y=y'. Find and prove a closed form expression for the number of Hamiltonian paths (paths visiting each vertex exactly once) from (1,1) to (n,3) in Gₙ.

Let M be a connected topological manifold (second countability assumed), and U⊂M a proper open subset. Show that there exists a subset A⊂U with empty interior such that every connected component of M-A contains exactly one connected component of M-U.

Bonus: What’s the largest generalization you can prove?