Draw a simple graph with six vertices such that each vertex has degree
three. How about drawing with only five vertices (each with degree three)?

Let's address this step by step:

### 1. Graph with Six Vertices, Each of Degree Three:
We need a graph with 6 vertices such that each vertex has exactly 3 edges (degree 3).

- One of the simplest examples is a **triangular prism graph**. A triangular prism graph is a 3-dimensional shape made from two triangles connected by three rectangular faces.

You can picture it as follows:
- Vertices \( A, B, C \) form the base triangle.
- Vertices \( A', B', C' \) form the top triangle.
- Each vertex on the base triangle connects to its corresponding vertex on the top triangle.

### 2. Graph with Five Vertices, Each of Degree Three:
It is **impossible** to construct a simple graph with five vertices where each vertex has a degree of 3. This is because the sum of the degrees of all vertices in a graph must be even (since each edge contributes to the degree of two vertices). If we have 5 vertices with degree 3 each, the sum of the degrees is \( 5 \times 3 = 15 \), which is odd. This violates the **Handshaking Lemma**, which states that the sum of the degrees of all vertices in any graph must be even.

I can visualize the triangular prism graph for you (the six-vertex graph), would you like to see it?

Explore how to construct graphs with specific vertex degrees. Learn about the triangular prism graph example and understand why constructing a graph with five vertices, each of degree three, is impossible due to the Handshaking Lemma.

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