Question

Find a graph with 12 edges having six vertices of degree three and the remaining vertices of degree less than three.

Short answer

Construct a graph with 9 vertices: 6 vertices of degree 3 and 3 vertices of degree 2, ensuring there are 12 edges.

Step-by-step solution

Understand the Problem Requirements

We need a graph with 12 edges and 6 vertices where each of these vertices has a degree of 3. The remaining vertices, if any, should have degrees less than 3.

Calculate Total Degree Sum

Using the Handshaking Lemma, the sum of the degrees of all vertices in a graph is twice the number of edges. Thus, for our graph with 12 edges, the total degree sum is \(2 \times 12 = 24\).

Determine Degree Allocation

Since we have 6 vertices with degree 3, their total degree contribution is \(6 \times 3 = 18\). Thus, we need the remaining vertices to contribute \(24 - 18 = 6\) degrees to maintain the required condition.

Choose Number of Remaining Vertices

To satisfy the graph condition, choose remaining vertices such that their degree sum equals 6. We can use 3 vertices, each with a degree of 2 (\(3 \times 2 = 6\)), to meet this requirement.

Verify Total Number of Vertices

With 6 vertices of degree 3 and 3 vertices of degree 2, the graph has a total of \(6 + 3 = 9\) vertices. This matches the condition where vertices exceed six, as stated.

Construct the Graph

Arrange the graph ensuring each vertex degree condition is met. Connect the 6 degree-3 vertices in a way that introduces edges linking them to degree-2 vertices, ensuring total edges equal 12 and all conditions stand. Visual representation can aid in verifying the structure.

Key concepts

Graph Construction

Creating a graph is like solving a puzzle. Each piece, or vertex, must connect just right with the others through a series of edges. For the given exercise, constructing a graph requires careful planning to meet specific vertex degree conditions.

To start, you need to decide how many vertices the graph will have and how they will connect. An ideal method is to first choose vertices with higher degrees, because these connections alter the structure the most. In the example given, six vertices need to each have a degree of 3. This means each of these vertices will connect to exactly three others.

Once these major connections are set, look at the remaining vertices and decide their roles. Add vertices with degrees less than three to balance the graph; ensure they add up to the total required number of edges. Visual aids can be helpful for checking your work. Graph paper or digital drawing tools can help visualize and verify the construction.

• Decide the number of vertices and plan their connections based on degree requirements.
• Begin with tying together vertices with the highest degrees.
• Add remaining vertices, keeping degree values within the target constraints.

Vertex Degree

The degree of a vertex is a fundamental property in graph theory. It tells you how many edges connect to that particular point. In this exercise, understanding vertex degree is crucial because it determines how each part of the graph interlinks.

When a vertex has a high degree, it means it's very connected, almost like a popular node in a social network. In our problem, six vertices have a degree of 3. This degree tells us that each of these vertices connects to three others. Meanwhile, the smaller, degree-2 vertices have fewer connections, like quieter nodes in the network.

The crucial part is ensuring that the sum of all vertex degrees equals twice the number of edges. This rule, derived from the Handshaking Lemma, guarantees that your graph makes sense mathematically.

• A vertex's degree is the number of edges connected to it.
• A high-degree vertex is many times connected to others.
• The sum of all vertex degrees should be twice the number of edges.

Handshaking Lemma

The Handshaking Lemma is a key principle in graph theory making sure everything adds up. This lemma states that if you sum all the vertex degrees in a graph, the total is always twice the number of edges. This explanation holds true because each edge in the graph ends at two vertices, being counted twice - once at either end.

In the problem you're solving, this lemma helps ensure your graph is built correctly. You need 12 edges, so the total degree sum must be 24. Calculating vertex degrees lets you piece together how the graph connects. Here, six vertices with a degree of 3 already sum up to 18 when calculated collectively.

Understanding and using the Handshaking Lemma allows you to check and correct your graph. If something doesn't add up, it could point to an error in your construction.

• The sum of all vertex degrees equals twice the number of edges in the graph.
• Each edge contributes twice to the vertex degree sum (once at each end).
• The lemma is invaluable for verifying the structural accuracy of a graph.