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: Let G = (V, E) be a planar graph such that |V |< 19. Show that if the length of the shortest cycle…

A: For a given graph G=V,E where V≤19 that shortest cycle in G is 5, then we need to prove that G has a…

Q: 4. Assume that k, t are positive integers with 2k+1 < 2t. Construct a simple (2k+1)-regular graph…

A: A graph with no loops and no parallel edges is called a simple graph.   In graph theory, a regular…

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: >("₂¹). Let G be a graph of order n and size m such that m> connected. Make a sketch. Show that G is

A: As per the question we are given a graph G with order n and size m and it is given that…

Q: prove that any connected graph G in which every edge is contained in a cycle cannot be disconnected…

A: Want to prove:  If G is a connected graph in whichevery edge 'e' is contained in a cycle then g-{e}…

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: 7. Prove : Every graph with radius r and girth with 2r +l is regular

A: To Determine :-  Every graph with radius r and girth with 2r +1 is regular.

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

Q: Suppose that a graph G has n vertices, and all vertices have degree greater than or equal to“,.…

A: Let G be a graph with n vertices. Given that all vertices have degree greater than or equal to n-12.…

Q: Let n € N with n > 2 and let G₁ = (V, E) be the graph with vertex set V = {₁, V2, ..., V2n} and edge…

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Q: complement

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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&lt;4 are planar. it is known that Peterson graph is not…

Q: For a given graph G having n vertices and e edges which is connected and has no cycles, which of the…

A: In the question it is provided that a graph G with n vertices and e edges is connected and has no…

Q: Prove that if  v0 and v1  are distinct vertices of a graph  G  = (V,E) and a path exists in G from…

A: Let v0 and v1 be distinct vertices of a graph G=V,E. Assume that there exists path in G from v0 to…

Q: simple path in G from v0 to v1

A:

Q: 2. Let G and H be a graphs. Prove that if G = H, then H= G.

A: Given: It is given that let G and H be a graphs , the objective is to prove that if and then

Q: Draw the dual to S4.

A: As per the question we are given a planner graph S4 and w have to draw its dual graph.

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…

Q: Show that if G is a simple graph with n vertices (where n is a positive integer) and each vertex has…

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

Q: Prove: For any graph r with six vertices ror r complementary contains C3

A:

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: a) Prove that, for all integers K > 5 P(K + 1,5) + P(K,5) = 5P(K,4) b) What is graph, Explain its…

A: We solve only one question at a time. Please submit questions one by one which you want to be…

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

A: Let G be a 2-connected graph. e and f are parallel edges in G. Let H= G\e We have to prove that if…

Q: Proof 10.5: Prove that if a connected planar graph has e edges and u vertices with v 23 and no…

A: We have to prove that if a connected planar graph has e edges and v vertices  with v≥3 and no cycles…

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: Which of the following four graphs on five vertices is not isomorphic to any of the other three?…

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Q: Prove or disprove: There exists some n > 1 such that there is a simple graph on n vertices, where…

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

A: Given: The statements regarding the bipartite graph.We need to check which of the given statements…

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: T/F. if T give reason, F give ex   2. Any 5-regular graph must have at least 5 perfect matchings

A: According to the given information, it is required to tell whether the statement is true or false.

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: Determine if the following statement is true or false. If it is true, then prove it. If it is false,…

A: As per the question, we are asked to determine the truth value of a given statement about the…

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…

Q: theorems- Let G be a graph-then XCG) 23 ifand only if Ġ has an odd cycle.

A:

Q: Prove that for any two simple connected joint graphs G and H, L(G) UL(H) L(GUH).

A: Introduction: A connected joint graph is a graph formed by connecting two graphs by a single edge.…