Problem 2: • Is the graph G below Eulerian? If "Yes", find an Eulerian Circuit staring at v₁- (mark the first edge as 1, second one as 2, etc -check the notes); if "No", please explain why. Is the graph G Hamiltonian? If "Yes", find a Hamiltonian cycle. If "No", explain why. • Adding a new vertex v to G and joining v to every odd vertex of G. Determine whether this new graph is Eulerian or Hamiltonian or both or neither, and explain why.
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- Which of the graph/s above contains an Euler Trail? Which of the graph/s above is/are Eulerian? Which of the graph/s above is/are Hamiltonian?Problem 7. Determine if a graph with the property specified below exists. If it exists, draw the graph and give its adjacency matrix; otherwise, prove that the graph doesn't exist. (a) A simple digraph with in-degrees 0, 1, 2, 2 and out-degrees 0, 1, 1, 3. (b) A simple digraph with in-degrees 0, 1, 1, 2 and out-degrees 0, 1, 1, 1.Which of the graphs has a Hamiltonian circuit? a. graphs 1 and 2 onlyb. graph 2 onlyc. graph 1 onlyd. graphs 2 and 3 onlye. graph 3 onlyf. graphs 1, 2, and 3
- Which of the following is false? A.) Hamiltonian cycle can be converted to a Hamiltonian path by removing one of its edge. B.) Every graph that contains a Hamiltonian cycle also contains a Hamiltonian path and vice versa is true. C.) There may exist more than one Hamiltonian paths and Hamiltonian cycle in a graph. D.) A connected graph has as Euler trail if and only if it has at most two vertices of odd degreeWelcome to Murphman’s Amusement Park! A guest of the park would like to start at the Entrance, walk along each of the midway streets (colored in grey) exactly once, and then leave at the Exit. Note that the railroad tracks are colored purple and are not midway streets. a.Draw a graph in the box above corresponding to this guest’s situation, and determine whether or not the graph has an Eulerian circuit, and if so, give one. If it does not, then explain why not. If the graph does not have an Eulerian circuit, determine whether it has an Eulerian path, and if it does, give one. If it does not, then explain why not. b. Is it possible for the guest to start at the Entrance, walk along all of the midway streets (colored in grey) exactly once, and then leave at the Exit?1. Draw the edges needed in order to make the following graph complete. 2. Find any Hamiltonian circuit on your complete graph. Give your answer as a list of vertices, starting and ending at the same vertex. Example: ABCA IMAGE BELOW PLEASE HELP
- Q1- What is the minimum distance between points C and F? Q2- Which of the following is a Hamiltonian Circuit for the given graph? Q3- What is the length of the Hamiltonian Circuit described in Q2? Q4- Which vertex in the given graph has the highest degree?Theorem 3.5 states the following: Let G be a loopless graph with at least three vertices, and no isolated vertices. Then G is 2-connected if and only if, for every pair {e, f} of edges of G, there is a cycle of G that contains both e and f.Theorem; If n ≥ 3 and if d(xk) ≤ k < ½*n → d(xn-k) ≥ n - k Then graph G = (X,E) contains a Hamiltonian cycle. Proof please !
- Given the graphs shown below, determine which graph is Hamiltonian and for such graph, find a Hamilton cycle:Identify the following graphs if they are Hamiltonian.FOR 1-3: Consider the following graphs: 1. Which of the graph/s above contains an Euler Trail? A. A and D B. B and C C. A, B, and C D. B, C, and D 2. Which of the graph/s above is/are Eulerian? A. None of the graphs B. Only B C. Only C D. B and C 3. Which of the graph/s above is/are Hamiltonian? A. A and B B. A and C C. A, B, and D D. A, C, and D FOR: 4-8: Consider the following graph: 4. What is the minimum distance between points C and F? A. 9 B. 10 C. 11 D. 12 5. Which of the following is a Hamiltonian Circuit for the given graph? A. AIBGCDEFHA B. DICBAIHGFED C. FIBAHGCDEF D. GFEIDCBAHG 6. What is the length of the Hamiltonian Circuit described in number 46? A. 35 B. 37 C. 39 D. 41 7. Which vertex in the given graph has the highest degree? A. Vertex C B. Vertex F C. Vertex H D. Vertex I 8. Which is referred to as an edge connecting the same vertex? A. Circuit B. Loop C. Path D. Repeated Edge 9. Which of the following statements is/are true? i. The vertices of K4 all have…