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From: segre@uicsl.UUCP
Newsgroups: net.ai
Subject: Re: Re: Four color... - (nf)
Message-ID: <5716@uiucdcs.UUCP>
Date: Fri, 17-Feb-84 22:44:07 EST
Article-I.D.: uiucdcs.5716
Posted: Fri Feb 17 22:44:07 1984
Date-Received: Mon, 20-Feb-84 07:08:13 EST
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#R:mit-eddi:-129000:uicsl:15500029:000:1585
uicsl!segre    Feb 17 10:05:00 1984

Not quite...perhaps this will help clear things up:

The Heawood Theorem (1890) gives an upper limit on the chromatic number for
graphs embeddable on surfaces of genus > 0 as follows:

chi(G) .LE. floor([7 + sqrt(1 + 48 gamma) ] / 2)

where chi(G) is the chromatic number of a graph G and gamma is the
genus of the surface. Note that surface refers to an orientable surface
(i.e. having two sides: rules out things like the Mobius strip).

The genus of a sphere is 0, that of a torus is 1. Note that planar
graphs are those embeddable on the sphere.

Although Heawood's theorem gives an upper bound, it doesn't tell us
that this is the best possible bound: to do so, we must first see that
the (minimum) genus (i.e. the surface of smallest genus on which the
given graph G is embeddable) of the complete graph on n vertices
(call it K-sub-n) is given by:

gamma(K-sub-n) = ceiling( [(n-3)(n-4)] / 12)

Proof of this is quite complex, involving 12 cases (one to show each
congruence class of K-sub-n is embeddable on a surface of this genus).

Therefore, we see that for surfaces of genus > 0 Heawood's theorem
gives the best bound on chromatic number. Thus the "colorability" of
higher order surfaces such as the torus and double-torus has been well
understood since 1890.

Note that if you plug in 0 for gamma in Heawood's theorem you get 4,
implying that graphs embeddable on the sphere (plane) are 4-colorable.
However, Heawood's proof explicitly calls for non-zero gamma --
which is why the 4 Color Theorem was still only conjecture after 1890.


Alberto Segre
uiucdcs!uicsl!segre