From: eppstein@euclid.ics.uci.edu
Subject: Re: K_{3,6} in torus?
Newsgroups: sci.math
Date: 3 Jun 98 00:14:24 GMT

gerry@mpce.mq.edu.au (Gerry Myerson) writes:
> Can you draw K_{3,6} in a torus?

Yes.  Since 3 and 6 are such nice hexagonal numbers, use a hexagonal torus
(glue together opposite sides of a regular hexagon).

Fill the hexagon with six 60-120 degree diamonds, meeting at the
hexagon's center, such that the long diagonal of each diamond runs from
the center to a corner of the hexagon.  The other two vertices of each
diamond are the centers of six small equilateral triangles into which
the hexagon could be divided.

The gluing causes the remaining region outside these six diamonds to
form three more diamonds, one on each hexagon edge.

These nine diamonds are the faces of an embedding of K_{3,6}.
--
David Eppstein		UC Irvine Dept. of Information & Computer Science
eppstein@ics.uci.edu	http://www.ics.uci.edu/~eppstein/
==============================================================================

From: Rick Decker <rdecker@hamilton.edu>
Newsgroups: sci.math
Subject: Re: K_{3,6} in torus?
Date: Wed, 03 Jun 1998 13:18:03 -0400

Gerry Myerson wrote:
> 
> Can you draw K_{3,6} in a torus?
> 
Yes.  Ringel's theorem on the genus of complete bipartite graphs
indicates that the genus of K{3, 6} is 1, so it can be done.
Here's one embedding, where the vertex sets are {a, b, c, d, e, f}
and {x, y, z}.  Rather than attempt an ASCII picture, I'll represent 
the embedding by rotations at each vertex:

x: (abcdef)
y: (adcfeb)
z: (afcbed)

a: (xzy)
b: (xyz)
c: (xzy)
d: (xyz)
e: (xzy)
f: (xyz)

This gives the expected 9 faces, namely

[xayb], [xbzc], [xcyd], [xdze], [xeyf], [xfza], [yczf], [yezb], [yazd]
> 
> This isn't homework, but I may assign it to my topology class if I can
> work out how to do it...
> 
Go for it--they'll learn a lot from their attempts.


Regards,

Rick

p.s. A picture isn't all that hard.  In a square with the edges
identified in the usual way to make a torus, place the vertices
like this:

         y     e     d

         f     x     c

         a     b     z

and then embed edges as required.

-----------------------------------------------------
Rick Decker                   rdecker@hamilton.edu
Department of Comp. Sci.      315-859-4785
Hamilton College
Clinton, NY  13323            =  !=  ==  (!)
-----------------------------------------------------
==============================================================================

From: nikl+sm000029@pchelwig1.mathematik.tu-muenchen.de (Gerhard Niklasch)
Newsgroups: sci.math
Subject: Re: K_{3,6} in torus?
Date: 3 Jun 1998 14:10:53 GMT

In article <6l2tru$c2m$1@nnrp1.dejanews.com>,
 Robin Chapman <rjc@maths.ex.ac.uk> writes:
|> On course this raises a more general question. What is the genus of K_{m,n}
|> (i.e., the least genus of an orientable surface into which it embeds).

For starters, see Arthur T. White, `Graphs, Groups and Surfaces',
2nd revised and enlarged ed., Amsterdam: North Holland 1984
(Mathematics Studies; 8).

Standard Disclaimer: My copy is at home.

Enjoy, Gerhard
-- 
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