Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Path: utzoo!watmath!clyde!burl!hou3c!hocda!houxm!ihnp4!inuxc!pur-ee!uiucdcs!uicsl!segre 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 Lines: 41 #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