Homework 3 copy

# Homework 3 copy - k contains at least k leaves(6 For two...

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HOMEWORK 3 8 PROBLEMS DUE: WEDNESDAY, FEBRUARY 16, 2011 (1) Show that for each n N the complete graph K n is a contraction of K n,n . (2) For n N , can K n be a contraction of K m,n if m < n ? (3) The complete tripartite graph K r,s,t consists of three disjoint sets of vertices (of sizes r,s and t ), with an edge joining two vertices if and only if they lie in diﬀerent sets. Draw K 2 , 2 , 2 . What is the number of edges of K 2 , 3 , 4 ? (4) There are exactly 11 unlabeled trees on seven vertices. Draw these eleven trees, making sure that no two are isomorphic. (5) Show that every tree containing a vertex of degree
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Unformatted text preview: k contains at least k leaves. (6) For two points in R 2 , P 1 = ( x 1 ,y 1 ) and P 2 = ( x 2 ,y 2 ), let d : R 2 × R 2 → R be given by d ( P 1 ,P 2 ) = | x 2-x 1 | + | y 2-y 1 | . Show that d is a metric on R 2 . (7) For all n ∈ N what is the eccentricity of each vertex of K n ? How many centers does K n have? (8) Draw all spanning trees of the graph G . • < < < < < < < < < < ± ± ± ± ± ± ± ± ± ± G : • • • ± ± ± ± ± ± ± ± ± ± < < < < < < < < < < • • • • ? ? ? ? ? ? ? ? ?...
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