Exercise 9.5: A tree with 'a verticcs is called graceful if it's verticcs can be labeled with the integers l, 2 , 'H. such that the absolute values of the difference of the labels of adjacent. vertices are all different. Draw and lahel two non-isonnlrphic graceful trees on 6 vertices. 10' Week 9 Assignment - Math 300 - Discrete l\\*Ia.theinatics Exercise 9.6: A forest is a. simple graph where each component is a. tree. Find all the non- isomorphic forests with ve vertices and two or more components. Exercise 9.7: For the following degree sequences (list of the degrees of the vert-ices in a graph) determine if there exists a bipartite simple graph with these degrees. If yes construct it, if not explain why it does not exist. (a) (6, 6, 4, 4, 4, 4, 4, 4, 2, 2, 2); (b) (4, 4, 4, 4, 4, 3, 3, 2, 2); (c) (5, 5, 5, 5, 5, 5, 4, 4, 4) Proof 9.1: Prove that. if there are two tliIl'ereut paths between vertiees 3: and y in a simple. graph, then there must he a cycle in the graph. Proof 9.2: Prove the statement: In any tree there is exactly one path from any vertex to any other vertex. (I-lint: Try contradiction. If there are two paths try to show there is a. cycle.) Proof 9.3: Prove that if a simple graph contains an odd length closed trail on > 1:1 > 62 > - ~ > m,- then the graph contains an odd length cycle