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 € 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 f to a subset of vertices.)

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