Hamiltonian Path - Induction Proof

In summary, the proof for #1 shows that for all n there is a Hamiltonian path in Kn, and for n+1 the same holds true assuming that Kn has a Hamilton path.
  • #1
brojesus111
39
0

Homework Statement


Let G be a graph.
1. Let n be a natural number. Use induction to show for all n >= 2 Kn has a Hamiltonian path.
2. Explain how you could use the proof from #1 to show that for all n (natural number) n > 2 Kn has a Hamiltonian cycle.

Homework Equations





The Attempt at a Solution



So Kn refers to a complete graph - I know that much. And the n refers to the number of vertices. I'm not sure how to begin the proof for #1.

For #2, the subgraph excluding one vertex is Kn-1 has a path. We can connect the ends together through the excluded vertex to make a cycle.
 
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  • #2
For part 1, can you show that K2 has a a Hamilton path?
Then, by induction, suppose Kn has a Hamilton path. You know that Kn is a subgraph of K(n+1); so what then can you say about the existence of a Hamilton path in this subgraph? Show that this can be extended to a Hamilton path in K(n+1). In fact, this path can be extended in precisely two ways.
 
  • #3
christoff said:
For part 1, can you show that K2 has a a Hamilton path?
Then, by induction, suppose Kn has a Hamilton path. You know that Kn is a subgraph of K(n+1); so what then can you say about the existence of a Hamilton path in this subgraph? Show that this can be extended to a Hamilton path in K(n+1). In fact, this path can be extended in precisely two ways.

So K2 has a Hamilton path is our base case, and that is obvious since it is just two vertices and we can easily visit each point once.
Our induction hypothesis is that we assume Kn has a Hamilton path.
For our inductive step, we want to show that this holds true for n+1. As you said, Kn is a subgraph of K(n+1). Since K(n+1) is a complete graph, then every vertex has a unique edge, so we can just travel along that edge to make that a Hamilton path since we assumed Kn has a Hamilton path.

I'm assuming I'm missing something here.
 
Last edited:
  • #4
No, you aren't missing anything; it's just a very easy problem.
 
  • #5
christoff said:
No, you aren't missing anything; it's just a very easy problem.

Oh, ok. Thanks for the help!
 

FAQ: Hamiltonian Path - Induction Proof

1. What is a Hamiltonian Path?

A Hamiltonian Path is a path in a graph that visits each vertex exactly once. It is a special case of a Hamiltonian Cycle, which is a cycle that visits each vertex exactly once and ends at the starting vertex.

2. What is an Induction Proof?

An Induction Proof is a mathematical technique used to prove that a statement is true for all natural numbers. It involves showing that the statement is true for a base case, and then using that base case to prove that the statement is also true for the next case. This process is repeated until the statement is proven to be true for all natural numbers.

3. How is Induction Proof used to prove the existence of a Hamiltonian Path?

To prove the existence of a Hamiltonian Path using Induction Proof, we first show that the statement is true for a base case, such as a graph with 3 vertices. Then, we assume the statement is true for a graph with n vertices and use this assumption to prove that the statement is also true for a graph with n+1 vertices. This process is repeated until we can prove that the statement is true for any graph with n vertices.

4. What are the key steps in an Induction Proof for a Hamiltonian Path?

The key steps in an Induction Proof for a Hamiltonian Path are:
1. Show that the statement is true for a base case, such as a graph with 3 vertices.
2. Assume the statement is true for a graph with n vertices.
3. Use this assumption to prove that the statement is also true for a graph with n+1 vertices.
4. Repeat this process until we can prove that the statement is true for any graph with n vertices.

5. What are some real-life applications of Hamiltonian Paths?

Hamiltonian Paths have many real-life applications, including:
1. In logistics, finding the most efficient route for a delivery truck to visit all locations.
2. In computer science, creating efficient algorithms for solving problems, such as the traveling salesman problem.
3. In chemistry, determining the shortest possible path for a chemical reaction to occur.
4. In network design, finding the most efficient way for a data packet to travel through a network.

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