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a) Suppose that T is a tree with n vertices, where n ≥ 1. Show that T has n − 1 edges. b) Conversely, suppose that G is a connected graph with n vertices and n − 1 edges, where n ≥ 1. Show that G must have at least one vertex of degree 1. Hence deduce that G must be a tree. c) Deduce that every connected graph with Euler characteristic 1 is a tree. d) Find an example of a graph with 5 vertices and 4 edges that is not a tree. 2. Consider the following polygon (otherwise kno ...

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