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Figure 819 The four-color problem
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show that any map whatever can be colored with ve colors Heawood showed that if the number of edges around each region in the map is divisible by three, then the map is four-colorable Heawood found a formula that gives an estimate for the chromatic number of any surface Here the chromatic number (g) of a surface is the least number of colors it will take to color any map on that surface We write the chromatic number as (g) In fact the formula is (g) 1 7+ 2 48g + 1
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so long as g 1 Here is how to read this formula It is known, thanks to work of Camille Jordan (1838 1922) and August M bius (1790 1868), that any surface in o space is a sphere with handles attached (see Fig 820) The number of handles is called the genus, and we denote it by g The Greek letter chi ( ) is the chromatic number of the surface the least number of colors that it will take to color any map on the surface Thus (g) is the number of colors that it will take to color any map on a surface that consists of the sphere with g handles Next, the symbols stand for the greatest integer function For example 9 = 4 just because the greatest integer in the number four and a half is 4 2 Also = 3 because = 314159 and the greatest integer in the number pi is 3
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Figure 820 The structure of a closed surface in space
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Figure 821 The torus is a sphere with one handle
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Now a sphere is a sphere with no handles, so g = 0 We may calculate that (g) 1 7 + 48 0 + 1 2 = 1 (8) = 4 2
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This is the four-color theorem! Unfortunately, Heawood s proof was only valid when the genus is at least 1 It gives no information about the sphere The torus (see Fig 821) is topologically equivalent to a sphere with one handle Thus the torus has genus g = 1 Then Heawood s formula gives the estimate 7 for the chromatic number And in fact we can give an example of a map on the torus that requires seven colors Here is what Fig 822 shows It is convenient to take a pair of scissors and cut the torus apart With one cut, the torus becomes a cylinder; with the second cut it becomes a rectangle
Figure 822 The torus as a rectangle with identi cations
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3 2 1 7 6 5 4 3 2 1
Figure 823 A map on the torus that requires seven colors
The arrows on the edges indicate that the left and right edges are to be identi ed (with the same orientation), and the upper and lower edges are to be identi ed (with the same orientation) We call our colors 1, 2, 3, 4, 5, 6, 7 The reader may verify that there are seven countries shown in our Fig 823, and every country is adjacent to (that is, touches) every other Thus they all must have different colors! This is a map on the torus that requires seven colors; it shows that Heawood s estimate is sharp for this surface Heawood was unable to decide whether the chromatic number of the sphere is 4 or 5 He was also unable to determine whether any of his estimates for the chromatic numbers of various surfaces of genus g 1 were sharp or accurate That is to say, for the torus (the closed surface of genus 1), Heawood s formula says that the chromatic number does not exceed 7 Is that in fact the best number Is there a map on the torus that really requires seven colors And for the torus with two handles (genus 2), Heawood s estimate gives an estimate of 8 Is that the best number Is there a map on the double torus that actually requires eight colors And so forth: we can ask the same question for every surface of every genus Heawood could not answer these questions
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