Let G=(V, E) be a graph such that • V| ≥ 3, G has exactly 2 leaves, and in G all the non-leaf vertices have degree 3 or more. ove that G has at least one cycle. You are not required to draw anything in your proof.

Q: Answer the following questions: a.Suppose that a connected bipartite planar simple graph has e…

A: "Hey ,since there are multiple questions posted, we will answer first question. If you want any…

Q: Exhibit a perfect matching in the graph below or give a short proof that it has none.…

A: Given,A graph.Aim: To exhibit a perfect matching in the graph given or give a short proof that it…

Q: 1a. Prove: every tree with n ≥ 2 vertices has at least 2 leaves. (3 pt) 1b. Let T be a tree. Prove:…

A: Question 1a:Let T be a tree with n-vertices, n≥2.  We have to  prove that the tree has at least 2…

Q: Let G = (V, E) be a connected graph. Use induction to show that if G contains exactly |G| = |V|.…

A: Suppose G=(V,E) be a connected graph, also let that G contains one cycle. Then we show that G=V

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Q: Prove that every graph G on {1, ... n} which is acyclic and has n – 2 edges has exactly 2 connected…

A: A graph with v vertices and e edges has at least v-e connected components. By induction on e, If e=0…

Q: The most famous theorem in graph theorem is the following: Theorem 1. The vertices of a planar graph…

A: No, it is not true that all graphs with χG<4 are planar. it is known that Peterson graph is not…

Q: (b) Give an example of a disconnected simple graph on n vertices in which every vertex has degree…

A: We need to give an example of a disconnected simple graph on n vertices in which every vertex has…

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A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: The graph of g con

A: Consider the graph below

Q: How can I prove "A graph is k-colourable if and only if it is k-partite.”?

A: As per the question we have to prove that a graph is k-colourable if and only if it is k-partite. To…

Q: 3. (Exercise 3.4) Prove Lemma 3.3: "Let G be a 2-connected graph. If e and f are parallel edges in…

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Q: can you please provide explanations

A: 1. Assumption: Start with the assumption that (G1​) and (G2​)are isomorphic graphs. 2. Definition of…

Q: If J has at least one vertex of degree greater than zero then there are two of even degree as long…

A: This is a problem of Graph Theory.

Q: Select all that are true. a) For any bipartite graph G = (LUR, E), if |L| = |R| then ElEL deg(l) =…

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Q: Turn in: 12.1 21 and 22 and the following: Consider the two vectors u = (2, 4) and 7 = (5, 1) shown…

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Q: b) Prove that G = K2,12 is planar by drawing G without any edge crossings. (c) Give an example of a…

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Q: Complete a path straight at vertex a that ends at vertex c.

A: A complete path is defined when all the vertices are passed through and none of the smaller paths…