hw7 - u and v have dierent colors. Let ( G ; k ) be the...

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MATH 145: HOMEWORK 7 ANDREW BERGET Problem A. Prove the Cayley-Pr¨ufer theorem: ( x 1 + x 2 + · · · + x n ) n - 2 = X T x d T (1) - 1 1 x d T (2) - 1 2 . . . x d T ( n ) - 1 n . Here the sum is over labeled trees T with vertex set [ n ] and d T ( i ) is the degree of vertex i in T . Problem B. Determine the number of spanning trees of the Petersen graph, shown below. Figure 1. The Petersen graph. Image from Wikipedia. Problem C. The following is a problem from your book: Prove that a connected graph G has at least | E ( G ) |-| V ( G ) | +1 cycles. Do this by inducting on the number of edges of G . Problem D. Determine a tree whose Pr¨ufer code is (3141593). Problem E. A k -coloring of a graph is an assignment of one of k different colors to each vertex of the graph such that if u and v are the end points of an edge, then
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Unformatted text preview: u and v have dierent colors. Let ( G ; k ) be the number of k-colorings of a graph G . Prove that ( G ; k ) = ( G-e ; 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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