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Explorations in Topology by David Gay

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12

Coloring Maps on Surfaces

Coloring maps on an island and on the sphere was the topic that began this book. This chapter brings back this topic but for maps on an arbitrary surface, fitting because we had just finished the complete classification of surfaces. The chapter is devoted to proving Heawood’s inequality which provides an upper bound for the number of colors needed to color a map on a surface based on the Euler number (assumed ≤0) of the surface. In 1890, Heawood claimed that this inequality was exact, that is, for each surface there exists a map requiring the number of colors given by the upper bound. The notes for the chapter, more extensive than usual, outline the history of subsequent attempts to prove Heawood’s claim. The claim ...

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