# Kuratowski theorem in Möbius band

I was curious to know where the theorem is failing (a graph is planar iff it doesn't contain subgraphs isomorphic to $K_{3,3}$ or $K_5$) Obviously, the theorem fails if we're in a Möbius band, as it can be seen here: http://gaussianos.com/el-problema-de-las-tres-casas-y-los-tres-suministros-y-la-banda-de-mobius/ (it's in spanish, but pictures speak for themselves)

So, that link talks about both cases $K_{3,3}$ and $K_5$, but I will only talk about the first one. The proof of the theorem only in this way: "If a graph contains a subgraph of $K_{3,3}$ then it's not planar" is simple:

If the graph was planar, it should be that $V-E+R=2$ (V: vertices, E: edges, R: regions), and so the number of regions would have to be $R=2+E-V=2+9-6=5$. But in the graph $K_{3,3}$, a cycle must have at least four edges, as it's bipartite, and so in the best case where all regions are bounded by four edges, we get a maximum number of regions of: $\frac {9}{4}2=4.5$, so $R\leq 4$. So it can't be planar.

We haven't talked here about the nature of the space we're dealing with, and we got to the conclusion only from Euler's formula, so that's what must be wrong. Now I go to the proof of that formula: I only prove it for graphs, by induction on $E$ starting with a segment ($V=2$, $E=1$, $R=1$), separating two cases:

Case 1: The graph is not a tree, then we can add 1 edge by ether adding one vertex and one edge to that vertex from any other vertex, so $E'=E+1$, $V'=V+1$ $R'=R$ (we add no regions), and if the formula was right for the previous state it is right now, as we're adding 1 and substracting 1. The other way is to add an edge between two vertices forming a new loop (dividing a region in two), and so we add no vertices, but we add one region and one edge, and so the induction step still works.

Case 2: The graph is a tree, it's actually the same, we can either add one new vertex and connect it, or add an edge to form a loop, so it stops being a tree.

That's more or less the proof I can find in Discrete math books.

The thing is that all inductions steps seem so obvious, that I can't see where it fails in a Möbius band. Actually the formula holds: $V-E+R=6-9+5=2$, so it's not this what doesn't hold. I don't know where the reasoning for Kuratowski's theorem is wrong.

I would appreciate some explanation (that doesn't involve really complex or advanced math)

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