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> **Possible Duplicate:**  
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> [The Klein bottle and the Heawood Conjecture](https://mathoverflow.net/questions/51830/the-klein-bottle-and-the-heawood-conjecture) 
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It is well known that the [Heawood Conjecture](http://mathworld.wolfram.com/HeawoodConjecture.html) states that the bound for the number of colours which are sufficient to colour a map on a surface of genus $g>0$ is, $$\gamma(g)=\left \lfloor \frac{7 + \sqrt{1+48g}}{2} \right \rfloor$$
as proved by the works of mostly Ringel, Young, and Gustin in the 1950s and 1960s. While two pathological cases were ruled out by the exhaustive proof of the Four Colour Theorem by Appel and Haken in the 1970s, namely the plane and the sphere, the Klein Bottle remains to defy that this condition is also necessary as shown by the [Franklin Graph](http://en.wikipedia.org/wiki/Franklin_graph).


Does anyone know of any papers that comment on the status of the Klein Bottle in this regard, in that there is some sort of explanation for why the Heawood Conjecture is true for absolutely every surface of genus $g$, except the Klein Bottle? I know that the Klein Bottle is spectacular in its non-orientable no-boundary way, but are there any known properties about the Klein Bottle that suggest a reason for why it defies the Heawood Conjecture? May it somehow relate to [fundamental polygons](http://en.wikipedia.org/wiki/Fundamental_polygon), [cross caps](http://mathworld.wolfram.com/Cross-Cap.html), or the [independence polynomial](http://mathworld.wolfram.com/IndependencePolynomial.html) of the Franklin Graph?


Any ideas, references to papers, expository articles, or original explanations would be welcome.



 Klein Bottle exception to the Heawood Conjecture
The derrière cri seems to be [this paper](http://people.math.gatech.edu/~thomas/PAP/5colkb.pdf) (in case the link goes away:
five coloring graphs on the Klein Bottle, by (wait for it):
Chenette, Postle, Streib, Thomas, Yerger.


