The problem you're asking about involves a simple graph $G$ with 12 edges and a set of conditions for the degrees of its vertices. Specifically, 6 vertices have degree 3, and the rest have a degree less than 3. You are asked to find the minimum number of vertices $G$ can have and draw the graph.

Solution Outline:

1. Graph Information:

   • Total number of edges: $12$.
   • There are 6 vertices with degree 3.
   • The remaining vertices have degrees less than 3.

2. Formula for the Sum of Degrees: The sum of the degrees of all vertices in a graph is equal to twice the number of edges. This is derived from the handshaking lemma, which states: $\text{Sum of degrees of vertices} = 2 \times \text{Number of edges}$ Substituting the number of edges: $\text{Sum of degrees} = 2 \times 12 = 24$

3. Degrees of 6 Vertices: Each of the 6 vertices has degree 3, so their total degree contribution is: $6 \times 3 = 18$

4. Remaining Degree Contribution: The remaining vertices contribute to the total degree sum. Let the number of these vertices be $n$, and each of them has a degree less than 3. Let’s assume each has degree 2 to minimize the number of vertices. The total degree contribution of these vertices is: $n \times 2$

   Combining the two contributions, we get: $18 + 2n = 24$ Solving for $n$: $2n = 6 \quad \Rightarrow \quad n = 3$

So, there are 6 vertices with degree 3, and 3 vertices with degree 2, making a total of $6 + 3 = 9$ vertices in the graph.

Conclusion:

The minimum number of vertices $G$ can have is 9.

Would you like help with drawing the graph or further explanations on the steps?

Here are 5 related questions to explore this concept further:

1. How do you prove that the sum of the degrees of the vertices in any graph is always even?
2. What is the handshaking lemma, and how is it applied in graph theory?
3. How would the graph change if the degree of the remaining vertices were less than 2?
4. What is the relationship between the number of vertices and edges in a connected graph?
5. How can you construct a simple graph with a given set of vertex degrees?

Tip: When working with vertex degrees, always ensure the sum of the degrees equals twice the number of edges—this is a quick way to check if the graph is possible.