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When working with connected planar graphs, understanding Euler's formula is crucial. Euler's formula relates the number of vertices (\textbf{V}), edges (\textbf{E}), and faces (\textbf{F}) in a planar graph. The formula is expressed as: \[ V - E + F = 2 \]
This relation helps in determining the unknown quantity when the other two are known. For instance, if we know the number of vertices and edges, we can easily calculate the number of faces (\textbf{regions} the plane is divided into). In our exercise, after determining that we have 6 vertices (V = 6) and 12 edges (E = 12), we use Euler's formula: \[ 6 - 12 + F = 2 \]
Simplifying this equation, we find: \[ F = 8 \] This tells us that the planar graph is divided into 8 regions.

The Handshaking Lemma is a fundamental theorem in graph theory. It states that the sum of the degrees of all vertices in a graph is equal to twice the number of edges. Mathematically, this is represented as: \[ \text{Sum of degrees} = 2 \times \text{Number of edges} \]
In our exercise, each of the 6 vertices has a degree of 4. So, the sum of the degrees is: \[ 6 \times 4 = 24 \]
According to the Handshaking Lemma, this sum equals twice the number of edges: \[ 24 = 2E \]
Solving for \textbf{E}, we find: \[ E = \frac{24}{2} = 12 \] Thus, the graph has 12 edges.

The degree of a vertex in a graph is the number of edges incident to it. In simpler terms, it's how many edges are connected to a specific vertex. For example, in our given exercise, each vertex in the planar graph has a degree of 4, meaning each vertex connects to exactly 4 edges.
Knowing the degree of each vertex helps in computing the total degree of the graph, which is useful when applying the Handshaking Lemma.

A planar graph can be drawn on a plane without any edges crossing. When such a graph is drawn, it divides the plane into distinct regions or faces. Each of these regions is enclosed by edges of the graph. Using Euler's formula, we can determine the number of these regions if we know the number of vertices and edges. In our exercise, after applying Euler's formula, we determined that the planar graph with 6 vertices, each of degree 4, and 12 edges, divides the plane into 8 regions.