Here is one:

To make this rectangular figure into torus, first imagine that the upper

and lower edges of this rectangle are bent toward each other until
they

touch.

Thomas L. Saaty - Paul C. Kainen: The Four-Color Problem.

Assaults and Conquest.

New York: Dover, 1986, p. 35.

Antreas P. Hatzipolakis <xpolakis@otenet.gr>

Another 7 color map on the torus may be found in Coxeter's

Introduction to Geometry, p. 387, Figure 21.3b. This is a nice

symmetric tessellation of the torus by seven hexagons, cut out

of the hexagon tessellation of the plane, and Coxeter attributes

it to Heawood.

Regards,

John Stillwell

Heawood's original paper, titled "Map-Colour Theorem," appeared in the

Quarterly Journal of Pure and Applied Mathematics, vol. 24 (1890),
pp.

332-338. The pertinent excerpt is reprinted in the book "Graph
Theory

1736-1936," by Biggs, Lloyd and Wilson (Oxford University Press, 1976),

pp. 112-115. There is an figure there of a torus map needing
7 colors.

I assume it's an exact reproduction of what's in the original paper.

Perhaps someone with access to the Quarterly Journal can confirm that.

My reason for assuming it's exact is that Biggs et al. also reprint

Kempe's 1879 "proof" of the 4-color theorem, which appeared in the

American Journal of Mathematics, and the figures there are identical

with those in a fascimile version of AJM available online through JSTOR

( http://www.jstor.org - but you need an institutional membership
to

do any searches). Again, perhaps someone with access to the original

can confirm that JSTOR's version is exact. (I'm pretty sure it
is. I

think what you see at JSTOR are scanned versions of the actual journals.)

In his 1879 paper, Kempe describes (in words) a map on a torus that

requires 6 colors. Presumably Heawood was the first person to
find one

that needs 7.

Barry Cipra

cipra@microassist.com

Heawood's original paper, titled "Map-Colour Theorem," appeared in the

Quarterly Journal of Pure and Applied Mathematics, vol. 24 (1890),
pp.

332-338. The pertinent excerpt is reprinted in the book "Graph
Theory

1736-1936," by Biggs, Lloyd and Wilson (Oxford University Press, 1976),

pp. 112-115. There is an figure there of a torus map needing
7 colors.

In case anyone simply wants a glimpse of the figure and the reference
to it

on p. 334 of Heawood's paper, I have scanned and placed them here:

http://www.lsus.edu/sc/math/rmabry/temp/heawood.htm

Rick Mabry