Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: $Revision: 1.6.2.14 $; site uiucdcs.UUCP Path: utzoo!watmath!clyde!burl!ulysses!mhuxl!ihnp4!inuxc!pur-ee!uiucdcs!west 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 References: <149@ihnet.UUCP> Lines: 18 Nf-ID: #R:ihnet:-14900:uiucdcs:28200041:000:837 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.