Draw a single graph G which satisfies all of the following properties. - All vertices of G are of degree 3, - G is connected, - the chromatic number of G is 3 - and G is planar.

Q: Consider the graph as shown in figure and determine the following: Pendant vertices, pendant edges,…

A: 1. Pendant vertices are d,e,f,g Pendant edges are ad edge,eb edge,fb edge,gb edge, Odd vertices are…

Q: 5. Let G = (V, E) be a finite, connected graph. Show that if no vertex of G has degree less than 2,…

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Q: Let G be a simple connected graph with exactly 8 vertices, one of them has degree 7, and the other 7…

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Q: Use Euler's theorem to determine whether the following graph has an Eul A connected graph with 78…

A: Euler theorem: A path that starts and ends at different vertices and passes across every edge of a…

Q: Consider the two following graphs and denote the upper graph as G and the lower graph as H. Answer…

A: Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: (d) Suppose that a graph G is regular of degree r, where r is odd. (i) Prove that G has an even…

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Q: Consider the 4-vertex graph G associated with the Seven Bridges of Königsberg problem from the…

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Q: Theorem 1: Suppose G is a connected graph. For every pair of vertices u and v in G, every uv-path…

A: To prove this theorem, first we will recall the definition of partite graph

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 G be a connected graph with n(G) ≥ 2. Show that G at least 2 vertices which are not cut-vertices…

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Q: Suppose a (simple) graph G = (V, E) is regular with odd vertex degree. Prove that the number of…

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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: .34. Consider the graph G in Fig. 8-58. Find: (a) degree of each vertex (and verify Theorem 8.1);

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Q: Let G be a planar graph in which every face is either a triangle or a pentagon. If G has 12…

A: It is given that Let G be a planar graph in which every face is either a triangle or a pentagon. If…

Q: Show that there is a graph G with exactly 5 nodes where both G and its complement, have chromatic…

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Q: Show that if u and v are the only odd-degree vertices in G, then there is a uv path G.

A: We prove this by induction on the number of vertices in G.   Case 1:   If the graph consists of…

Q: G is a graph on n 21 vertices such that both G and its complement have an Euler trail. (Recall that…

A: To do: Draw a simple graph G such that "G is a graph with n≥1 vertices such that both G and its…

Q: Give an example of a graph with an infinite number of vertices that has a finite chromatic number.…

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Q: 4. Give an example (if it exists) of each of the following: (a) a bipartite graph that is regular of…

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Q: Let G be a connected graph that has an Euler tour. Prove or disprove the following statements. If G…

A: Euler Circuit: A closed walk in a graph G that uses every edge exactly once is called an Euler…

Q: Which of the following statements about a connected graph is always TRUE?         A path of…

A: Solution: From the definition of the connected graph there should be path between any two pair of…

Q: 2. Prove the following and make a sketch. a) Prove that every Eulerian graph of odd order has three…

A: 1. Since G is connected, every vertex has positive degree. Since G has an Eulerian cycle, every…

Q: (b) Show that G may not necessarily have three cycle subgraphs by giving an example of a graph G,…

A: Consider the equation

Q: Find a subgraph H of the graph G given below that has 4 vertices and 3 edges.

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Q: In each part, draw a (simple) graph G with the given properties or argue that no such graph exists.…

A: As per the question we have to draw a connected graph having 6 vertices and 5 articulation points.

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 graphs are bipartite? I

A: A graph is bipartite iff graph does not contains any odd cycle.

Q: Prove that if G is a simple graph with n > 3 vertices and (",') +2 edges, then G is Hamiltonian. b)

A: In the given question we have to prove that a simple graph with vertices more than or equal to three…

Q: 2. Prove that the size of every k-chromatic graph is at least (

A: Consider the provided question, According to you we need to solve only (2). (2) We need to prove…

Q: State whether the following statement is true or false. Justify your answer with a short proof or a…

A: Consider the general case, Let G be k-edge connected graph then we shall rove that G is k connected.…

Q: If G is a simple graph with at least two vertices. prove that G must contain two or more vertices of…

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Q: 5. For any graph G with six vertices, then G or its complement Ġ contains a triangle.

A: The solution for the above question is as shown below.

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

A: This is the problem of graph theory.

Q: Let G be a graph of order 21 and size 50. There are three vertices of degree 2, six vertices of…

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Q: (b) Show that if a 2-colorable graph G has exactly two 2-colorings, then any two cycles contained in…

A: Given: Demonstrate that if a 2-colorable graph G has exactly two 2-colorings, then any two cycles in…

Q: A. If a simple graph G on 30 vertices has at least 301 edges, then its chromatic number is at least…

A: Note: As per bartleby instruction when more then one question is given only one has to be answered.…

Q: The number of non-isomorphic simple graphs on 8 vertices, all of which have degree 2, is Select one:…

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Q: Let G be a graph with exactly 10 vertices and 27 edges. Suppose that each vertex has degree 3, 5, or…

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Q: Let G be a simple connected graph with exactly 8 vertices, one of them has degree 7, and the other 7…

A: Consider G be a simple connected graph with exactly 8 vertices, one of them has degree 7, and…

Q: Let Gbe a simple planar graph containing no triangles. (i) Using Euler's formula, show that G…

A: Let G be a simple planar graph containing no triangles.

Q: Draw graph G and its complement, showing that at least one of G and it's complement, G', is…

A: Suppose G is disconnected. We want to show that G¯ is connected. So suppose v and w are vertices. If…

Q: Let G=(V, E) be a graph such that • V| ≥ 3, G has exactly 2 leaves, and in G all the non-leaf…

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Q: Show all possible connected sub graphs of the following graph. How many of them are there refer to…

A: Given: a graph

Q: Let G be a simple graph with 20 vertices. Suppose that G h ponents, and every pair of distinct…

A: Given: Simple graph G with 20 vertices. G has two-component with distinct vertices that satisfy…

Q: Prove that if G is a graph of order p and has size greater than ¹) then G is connected. 2

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Q: Let G be graph with 8 vertices and 10 edges. If every vertex of G has degree 2 or 3, then How many…

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Q: For any connected graph G of order 54, if x is a vertex of G, then the number of components of G-x…

A: Given: To explain the given statement as follows,

Q: Let G be a simple graph with nonadjacent vertices v and w, and let G+e denote the simple graph…

A: Given that, v and w are non-adjacent vertices in G. (1): G+e the simple graph formed by adding the…