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I just gave this problem on my graph theory final, and I can't wait to see what innovative approaches my students come up with.

The two ladders are defined to be the only $3$-regular graphs that can be obtained by adding two new edges to the $2\times n$ grid, where $n\ge 3$. (The names suggest which ladder is which.)

So the problem is to prove that ${\mathcal T}_n$ (toroidal ladder) is not isomorphic to ${\mathcal M}_n$ (Möbius ladder) for each $n\ge 3$.

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For even $n$ the toroidal ladder is bipartite, the Möbius ladder is not. For odd $n$ the Möbius ladder is bipartite, the toroidal ladder is not.

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  • $\begingroup$ Exactly. This is the simplest and most direct solution (also the solution I came up with). Nicely done. $\endgroup$
    – Doc
    Dec 15, 2013 at 15:24
  • $\begingroup$ Thanks. It is a nice problem, not too hard to solve, but not immediately obvious either. Please report elegant solutions by your students (if they are different). $\endgroup$ Dec 15, 2013 at 15:55
  • $\begingroup$ Also, one is planar and the other is not. $\endgroup$
    – hbm
    Dec 16, 2013 at 3:26
  • $\begingroup$ @hbm Absolutely, and this was the most common response I received among students who got at least partial credit. Few were actually able to prove this however. Many just stated it. $\endgroup$
    – Doc
    Dec 19, 2013 at 19:38

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