3.1.11
In Exercises 3.1.3 through 3.1.12, either draw the required graph or explain why no such graph exists.
3.1.11 A 10-vertex, 3-component, forest with exactly nine edges.
3.1.14
3.1.14 Prove that if \( G \) is a tree having an even number of edges, then \( G \) must contain at least one vertex having even degree.
3.1.19
3.1.19 Prove or disprove: There does not exist a connected \( n \)-vertex simple graph with \( n+2 \) edges that contains four edge-disjoint cycles.
3.1.29
3.1.29 Let \( T \) be a tree with at least two vertices. Prove that the center \( Z(T) \) is a single edge if and only \( \operatorname{diam}(T)=2 \operatorname{rad}(T)-1 \)
