Please answer 3 and 4 116.1012231. Consider the graph above. Give an ordering of the vertices soQuestion 5when you apply the greedy algorithm to color the tree when each new vertexis colored it is connected to exactly one vertex that was colored previously.2. What number of colors does the greedy algorithm use when applied to theordering from Part 1?3. Give an ordering of the vertices of the graph above, so that applying thegreedy algorithm uses more colors than in Part 2. What is the first vertexin your ordering which uses a color that was not used in Part 2?4. Let T be an arbitrary tree. Prove that there is an ordering on the verticesas in Part 1.

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Description: Please answer 3 and 4 116.1012231. Consider the graph above. Give an ordering of the vertices soQuestion 5when you apply the greedy algorithm to color the tree when each new vertexis colored it is connected to exactly one vertex that was colored prev

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3. Give an ordering of the vertices of the graph above, so that applying the greedy algorithm uses more colors than in Part 2. What is the first vertex in your ordering which uses a color that was not used in Part 2? 4. Let T be an arbitrary tree. Prove that there is an ordering on the vertices as in Part 1.

3. Give an ordering of the vertices of the graph above, so that applying the greedy algorithm uses more colors than in Part 2. What is the first vertex in your ordering which uses a color that was not used in Part 2? 4. Let T be an arbitrary tree. Prove that there is an ordering on the vertices as in Part 1.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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The following table presents the implementation of Dijkstra's algorithm on the evaluated graph G with 8 vertices. How many different shortest paths exist in the graph G between the vertices a and g?
1. Consider the graph above. Give an ordering of the vertices sowhen you apply the greedy algorithm to color the tree when each new vertexis colored it is connected to exactly one vertex that was colored previously.2. What number of colors does the greedy algorithm use when applied to theordering from Part 1?3. Give an ordering of the vertices of the graph above, so that applying thegreedy algorithm uses more colors than in Part 2. What is the first vertexin your ordering which uses a color that was not used in Part 2?4. Let T be an arbitrary tree. Prove that there is an ordering on the verticesas in Part 1.
Recall that: 1. a trail is a walk without repeated edges, 2. an Eulerian trail is a trail passing through every edge, and 3. a graph admits an Eulerian trail if and only if the following hold (a) every vertex has even degree or there are exactly two vertices with odd degree and (b) the vertices of nonzero degree are connected.Note that the following algorithm may be used to construct Eulerian trails. • Step 1. – If every vertex has even degree then pick any vertex v of nonzero degree and construct a circuit from v to itself. – If there are exactly two vertices with odd degrees then construct a trail between them. • Step 2. As long as some vertex w belongs to the current trail and has incident edges not in the current trail, construct another circuit from w to itself using only previously unused edge and join it to the current circuit. ANSWER THE FOLLOWING:1. Does the graph below admit an Eulerian trail. If so, find one. If not, prove it. 2. Prove that Kn, m admits a Eulerian circuit if…
Recall that: 1. a trail is a walk without repeated edges, 2. an Eulerian trail is a trail passing through every edge, and 3. a graph admits an Eulerian trail if and only if the following hold (a) every vertex has even degree or there are exactly two vertices with odd degree and (b) the vertices of nonzero degree are connected.Note that the following algorithm may be used to construct Eulerian trails. • Step 1. – If every vertex has even degree then pick any vertex v of nonzero degree and construct a circuit from v to itself. – If there are exactly two vertices with odd degrees then construct a trail between them. • Step 2. As long as some vertex w belongs to the current trail and has incident edges not in the current trail, construct another circuit from w to itself using only previously unused edge and join it to the current circuit. ANSWER THE FOLLOWING: - Prove that Kn, m admits a Eulerian circuit if and only if both n and m are even.
Let G = (X ∪ Y, E) be a bipartite graph such that the vertices are partitioned into two groups Xand Y , and each edge has one end point in X and one end point in Y . A 2-1 generalized matching is a set of edges S ⊂ E satisfying the following two conditions: 1. Every vertex in X belongs to at most two edges in S.2. Every vertex in Y belongs to at most one edge in S. How do I give an algorithm to find the size (number of edges) of maximum 2-1 generalized matching?
1)This problem will explore how different algorithms play out with the same weighted graph. a) Use the nearest neighbor algorithm to find an approximate solution to the traveling salesman problem for a circuit starting at vertex E, and find the weight of this circuit. b) Use the greedy algorithm to find an approximate solution to the traveling salesman problem for a circuit starting at vertex E, and find the weight of this circuit. Also, write down the edges in the order that you choose them when constructing the circuit as you apply the Greedy Algorithm. c) It turns out that neither the nearest neighbor algorithm nor the greedy algorithm gives a Hamiltonian circuit of minimum total weight starting at vertex E. Find some Hamiltonian circuit starting at vertex E of total weight smaller than those of the circuits you found in parts (a) and (b).
Find the nearest neighbor algorithm to find a Hamilton circuit that begins at vertex E.
Which graph has and Euler path? Euler Circuit?(Use Fleury’s Algorithm) For those that don’t give an explanation?
1. How many unique (excluding reverse) Hamiltonian circuits are possible? 2. Apply the Brute Force algorithm to find the minimum cost Hamiltonian circuit on the graph. What is the optimal circuit starting with vertex A, then B,...? 3. What is the optimal circuit starting with vertex A, then C,...? 4. The minimum cost is . IMAGE BELOW