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In graph theory, the Heawood conjecture or Ringel–Youngs theorem gives a lower bound for the number of colors that are necessary for graph coloring on a surface of a given genus:

I've learned that,

it takes $$\left\lfloor \frac{7 + \sqrt{49 - 24 \chi}}{2} \right\rfloor$$ colors except for the Klein bottle where it takes $6$, where $\chi$ is the Euler characteristic of the surface.

But the thing is that I'm interested in graphs on surfaces other than the plane that are still $4$-colorable. How could an example look like? Or does $4$-colorability imply that it can be drawn in the plane?

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draks ...
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  • It's pretty obvious that any graph that can be drawn on the plane can also be drawn on any other surface, so there are plenty of $4$-colourable graphs on say the Klein bottle. – Chris Eagle Aug 01 '13 at 20:44
  • Or do you mean you want a $4$-colourable graph that can't be drawn on the plane, and the stuff about other surfaces is irrelevant? – Chris Eagle Aug 01 '13 at 20:44
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    Any surface that forbids $K_5$ minors will force all graphs to be at most $4$-colorable. Any surface that doesn't forbid $K_5$ minors will allow $K_5$ to be drawn, which is $5$-colorable. Any graph with a $K_{3,3}$ minor without a $K_5$ minor will be $4$-colorable (at most) and cannot be drawn on a plane. – A.S Aug 01 '13 at 20:45
  • If the latter, then bisecting every edge of any graph gives you a topologically equivalent graph that's $2$-colourable. – Chris Eagle Aug 01 '13 at 20:47
  • @chris I mean graphs that can't be drawn on the plane. – draks ... Aug 02 '13 at 05:40
  • @andrew, this sounds like I'm looking for graphs with a $K_{3,3}$ minor on a surface that forbids $K_5$ minors. Do you know some example or tell how to get them? – draks ... Aug 02 '13 at 05:43

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Here's how you get a $3$-colourable map on the torus: Wrap up a chessboard to a torus and replace the vertices with small faces (just to make the map regular). With an odd (e.g. $9\times 9$) chessboard, the same construction gives you a $3$-colourable graph of the projective plane and from a $9\times 8$ (or other even/odd combination) you get a $3$-colourable map on the Klein bottle. These maps are not planar (unless the chessboard was too small) and give examples of maps that are even $3$-colourable. Some suitable modification of the maps is easily made to make a fourth colour necessary.

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