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Unformatted text preview: u and v have dierent colors. Let ( G ; k ) be the number of kcolorings of a graph G . Prove that ( G ; k ) = ( Ge ; k ) ( G/e ; k ) . Conclude that ( G ; k ) is a polynomial in k . (There is a subtlety involving multiple edges and loops. Obviously a graph with loops has no proper colorings. Multiple edges, on the other hand, have no eect on colorings. However, contracting and multiple edge should produce a loop in the contracted graph. To avoid this diculty, assume that G is simple and if you get multiple edges upon contracting an edge e , then you replace them with a single edge.) 1...
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 Winter '07
 Peche
 Math, Combinatorics

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