Torae Torae Torae
Bob underwood
that it would take no more than seven colors to "map a torus. this
was proved well before the proof that four colors would suffice to "map"
a plane. Now hat its been determined that a genus one torus can be
formed with a minimum of seven planes ,has it been proved that for every
color maping there is a corresponding planar maping for anymanifold?
___Bob U.---
John Conway
Please learn how to spell "tori"! I think you meant to ask whether
there was a similar theorem giving the highest number of colors that
could be needed for a map on an arbitrary 2-dimensional manifold.
Long long ago, Percy Heawood proved, except when K = 2, that any map
on a surface of Euler characteristic K could be colored with at most
f(K) colors, where I think f(K) = [K + root(49-24K)]. [The result is
true for K = 2, but Heawood didn't prove it.] In the thirties,
Franklin showed that for the Klein bottle, 6 colors suffice, so that the
Heawood bound can be improved in that case. I think it was nearly four
decades ago that Ringel and Youngs proved Heawood's bound was achieved
in all other cases.
John Conway
John Conway
On Tue, 19 Jan 1999, John Conway wrote:
>
> Long long ago, Percy Heawood proved, except when K = 2, that any map
> on a surface of Euler characteristic K could be colored with at most
> f(K) colors, where I think f(K) = [K + root(49-24K)]. [The result is
> true for K = 2, but Heawood didn't prove it.]
I see that with the function I gave, the result ISN'T true for K = 2,
so obviously I got it wrong. I'm obviously going to have to learn this
function by heart, since I can never reconstruct it!
John Conway