To determine whether it is possible to draw a connected graph in the plane with $8$ vertices, $10$ edges, and $5$ faces, we can use Euler's formula for planar graphs:

$V - E + F = 2$

where:

• $V$ is the number of vertices,
• $E$ is the number of edges, and
• $F$ is the number of faces.

Applying Euler's Formula

We have:

• $V = 8$ vertices,
• $E = 10$ edges, and
• $F = 5$ faces.

Plugging these values into Euler's formula:

$8 - 10 + 5 = 3$

However, for any connected planar graph, Euler's formula states that $V - E + F$ should equal $2$, not $3$. Since $3 \neq 2$, it is not possible to have a connected planar graph with $8$ vertices, $10$ edges, and $5$ faces.

Conclusion

No, it is not possible to draw a connected graph in the plane with $8$ vertices, $10$ edges, and $5$ faces, because it violates Euler's formula for planar graphs.

Would you like a deeper exploration of this concept or related examples?

Here are some follow-up questions to deepen understanding:

1. What types of graphs satisfy Euler's formula exactly?
2. How does Euler's formula apply to different types of surfaces, such as a torus?
3. Can a graph with more than 2 edges per vertex (non-planar) satisfy Euler's formula in the plane?
4. What are examples of graphs that satisfy $V - E + F = 2$ with different values for $V$, $E$, and $F$?
5. How does adding or removing edges affect the count of faces in planar graphs?

Tip: When solving graph theory problems, always verify basic constraints like Euler's formula first to determine the feasibility of the graph structure.