### Hypercubes in Different DimensionsA $ k $-dimensional hypercube on $ 2^k $ vertices is defined recursively. The base case, a 1-dimensional hypercube, is the line segment graph. Each higher-dimensional hypercube is constructed by taking two copies of the previous hypercube and using edges to connect the corresponding vertices. These connecting edges are shown in gray.#### Hypercubes:1. **1D Hypercube**: - A simple line segment connecting two points. - Represented by a pair of vertices with a single edge between them. - Diagram: ``` ●──● ```2. **2D Hypercube**: - Created by taking two 1D hypercubes and connecting the corresponding vertices. - Represented as a square with vertices connected at the corners. - Diagram: ``` ●──● │ │ ●──● ```3. **3D Hypercube**: - Created by taking two 2D hypercubes and connecting the corresponding vertices. - Represented as a cube with additional edges connecting vertices between the two squares. - Diagram: ``` ●──● /│ /│ ●──● │ │ ●─● │/ / ●──● ```### Proof Task**a. Prove that every $ k $-dimensional hypercube has a Hamiltonian circuit (use induction).**### Detailed Explanation- **1D Hypercube (Line Segment)**: - The simplest form, with only two vertices and one edge. trivially forms a Hamiltonian circuit.- **2D Hypercube (Square)**: - Two 1D hypercubes connected by four edges. The circuit can be visualized as moving along the perimeter of the square.- **3D Hypercube (Cube)**: - Two 2D squares connected by edges. The Hamiltonian circuit can be visualized by starting at one vertex and traversing the cube in a manner that each vertex is visited exactly once before returning to the starting vertex.Using induction, you can prove the Hamiltonian property for all higher dimensions by considering the construction of the $ k $-dimensional hypercube from two $(k-1)$-dimensional hypercubes.---This section is designed to help learners understand the

The given diagram: Hamiltonian circuit: It is a circuit that visits every vertex…

A k-dimensional hypercube on 2^k vertices is defined recursively. The base case, a 1- dimensional hypercube, is the line segment graph. Each higher dimensional hypercube is constructed by taking two copies of the previous hypercube and using edges to connect the corresponding vertices (these edges are shown in gray). Here are the first three hypercubes: O-O 1D: 2D: 3D: a. Prove that every k-dimensional hypercube has a Hamiltonian circuit (use induction).

A k-dimensional hypercube on 2^k vertices is defined recursively. The base case, a 1- dimensional hypercube, is the line segment graph. Each higher dimensional hypercube is constructed by taking two copies of the previous hypercube and using edges to connect the corresponding vertices (these edges are shown in gray). Here are the first three hypercubes: O-O 1D: 2D: 3D: a. Prove that every k-dimensional hypercube has a Hamiltonian circuit (use induction).

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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Question

### Hypercubes in Different Dimensions

A $ k $-dimensional hypercube on $ 2^k $ vertices is defined recursively. The base case, a 1-dimensional hypercube, is the line segment graph. Each higher-dimensional hypercube is constructed by taking two copies of the previous hypercube and using edges to connect the corresponding vertices. These connecting edges are shown in gray.

#### Hypercubes:
1. **1D Hypercube**:
- A simple line segment connecting two points.
- Represented by a pair of vertices with a single edge between them.
- Diagram:
```
●──●
```

2. **2D Hypercube**:
- Created by taking two 1D hypercubes and connecting the corresponding vertices.
- Represented as a square with vertices connected at the corners.
- Diagram:
```
●──●
│ │
●──●
```

3. **3D Hypercube**:
- Created by taking two 2D hypercubes and connecting the corresponding vertices.
- Represented as a cube with additional edges connecting vertices between the two squares.
- Diagram:
```
●──●
/│ /│
●──● │
│ ●─●
│/ /
●──●
```

### Proof Task

**a. Prove that every $ k $-dimensional hypercube has a Hamiltonian circuit (use induction).**

### Detailed Explanation

- **1D Hypercube (Line Segment)**:
- The simplest form, with only two vertices and one edge. trivially forms a Hamiltonian circuit.

- **2D Hypercube (Square)**:
- Two 1D hypercubes connected by four edges. The circuit can be visualized as moving along the perimeter of the square.

- **3D Hypercube (Cube)**:
- Two 2D squares connected by edges. The Hamiltonian circuit can be visualized by starting at one vertex and traversing the cube in a manner that each vertex is visited exactly once before returning to the starting vertex.

Using induction, you can prove the Hamiltonian property for all higher dimensions by considering the construction of the $ k $-dimensional hypercube from two $(k-1)$-dimensional hypercubes.

---
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