To show that ( u , v ) belongs to some minimum spanning tree of G .

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Chapter 23.1, Problem 1E

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To show that (u,v) belongs to some minimum spanning tree of G .

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Problem 3. Let (u, v) be a minimum-weight edge in a connected graph G. Show that (u, v) belongs to some minimum spanning tree of G.

(a) Let Ge (v, E) be an undinected let each edge e e E have weight weightle), and suppese all edge weights Then, the edge of minimum weight in connected graph, IVI >3 and are diffesent. Let T be a minimum spanning tree in. G. G must belang to T. an undirected graph G=(V,E), IVI ン3, has connect are minimum spanning tree, then all the edge weights unique よ5ferent. FOLLOWING STATEMET PROOF THE Oe SHOW A COUNTEREXAMPLE

1. Let G be a tree and let L be the set of leaves in G (the vertices of degree 1). (a) If G contains a vertex of degree k, show that |L| > k. (b) Let f be a graph isomorphism from G to G. Prove that f(L) = L. (c) Prove that either there is a vertex v e V(G) such that f(v) = v or there is an edge {x, y} € E(G) such that {f(x), f(y)} = {x, y}. (Hint: Induction on |V(G)|; in the induction step consider a restriction of ƒ to a subset of vertices.)