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From: bobm@rtech.UUCP (Bob Mcqueer)
Newsgroups: net.math
Subject: Re: Re: Euler's(?) formula
Message-ID: <513@rtech.UUCP>
Date: Sun, 23-Jun-85 19:07:59 EDT
Article-I.D.: rtech.513
Posted: Sun Jun 23 19:07:59 1985
Date-Received: Sat, 29-Jun-85 00:37:30 EDT
References: <673@lsuc.UUCP>
Organization: Relational Technology, Alameda CA
Lines: 27

> .....  Actually I recall that
> there's a simple extension of the formula that even includes those...
> I think F + V = E + 4T + 2 where T is the number of tunnels. .....
> 

The extension of Euler's formula for graphs with simply connected faces
embedded on spheres with n handles is:

V - E + F = 2 - 2N.

There is an intuitive proof of this one about on the level of the
"induction on the number of edges" proof for the original formula which
a couple people have referred to thus far.  The basic idea is that you
use the "simply connected" condition to assure that >= 1 arcs of the graph
run around each handle, then imagine cutting all the handles, creating
new edges and vertices around the circle of the cut.  Then pull the handles
back into the sphere, filling up the empty holes with new faces.  You added
the same number of new vertices and edges around each cut, and 2N new faces.
The result is a graph on a sphere, for which the original Euler formula
applies.  An illustration of this is given in a nice little elementary
text "Intuitive Concepts in Elementary Topology", by B. H. Arnold,
Prentice-Hall, 1962.

Boy, does this discussion take me back a long ways.

Bob McQueer
ihnp4!amdahl!rtech!bobm