Math Problem Statement

Apply the sorted edges algorithm to the graph. Give your answer as a list of vertices, starting and ending at vertex A.

Solution

To solve this problem using the sorted edges algorithm for finding a Hamiltonian Circuit, we need to follow these steps:

Step-by-Step Process:

  1. List all edges in ascending order of weights:

    • (A, C) = 2
    • (C, D) = 5
    • (D, E) = 6
    • (A, E) = 4
    • (B, C) = 9
    • (A, B) = 10
    • (B, D) = 12
    • (A, D) = 14
    • (B, E) = 13
    • (C, E) = 15
  2. Start adding edges from the list, without forming a smaller circuit or exceeding the degree of 2 for any vertex.

    • Add edge (A, C) = 2.
    • Add edge (C, D) = 5.
    • Add edge (A, E) = 4.
    • Add edge (D, E) = 6.
    • Add edge (B, C) = 9.

    At this point, all vertices are connected and a Hamiltonian Circuit is formed. The remaining edges would either exceed the degree of 2 for some vertices or create smaller loops.

  3. List the circuit: A possible Hamiltonian circuit starting and ending at vertex A could be:

    • A → C → B → D → E → A.

Would you like a deeper explanation of any of the steps or any further clarifications?


Here are 5 related questions to explore the topic further:

  1. What is a Hamiltonian path, and how does it differ from a Hamiltonian circuit?
  2. How does the sorted edges algorithm compare to other algorithms for finding Hamiltonian circuits?
  3. Can all graphs have a Hamiltonian circuit?
  4. What are some real-world applications of Hamiltonian circuits?
  5. How does the degree of a vertex affect the possibility of forming a Hamiltonian circuit?

Tip: When using the sorted edges algorithm, always ensure you don't close a smaller cycle prematurely or exceed the degree of 2 for any vertex before all vertices are included.

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Math Problem Analysis

Mathematical Concepts

Graph Theory
Hamiltonian Circuits
Sorted Edges Algorithm

Formulas

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Theorems

Hamiltonian Circuit Theorem

Suitable Grade Level

Grades 10-12