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From: west@uiucdcs.UUCP
Newsgroups: net.math
Subject: Re: coloring a 3-dimensional map - (nf)
Message-ID: <28200041@uiucdcs.UUCP>
Date: Mon, 20-Aug-84 23:19:00 EDT
Article-I.D.: uiucdcs.28200041
Posted: Mon Aug 20 23:19:00 1984
Date-Received: Tue, 21-Aug-84 07:17:06 EDT
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Nf-From: uiucdcs!west    Aug 20 22:19:00 1984

#R:ihnet:-14900:uiucdcs:28200041:000:837
uiucdcs!west    Aug 20 22:19:00 1984

The computer was used to aid in the search for a proof
of the four color ``conjecture'', by selecting which cases to analyze.
The resulting set of configurations has been checked by humans,
and the proof is accepted by most graph theorists.
What more do you want?

The answer to the base note is correct, since it is easy to
embed any 1-dimensional complex (i.e., graph) in 3-dimensional space.
The appropriate generalization of the 4-color theorem is to
surfaces of higher genus, not spaces of higher dimension.
The resulting formula is Heawood's Theorem.  For example,
any graph drawn on the torus (i.e., doughnut) can be colored with 7 colors,
and this is best possible.  Curiously, for the higher genus the upper
bound was the easy part, while showing that some graph with that
chromatic number embeds on that surface took 80 years.