Final answer:
According to Euler's formula, a connected graph with 7 vertices and 10 edges cannot be drawn without edges crossing. A graph with 10 vertices and 8 edges can be a connected planar graph. It is not possible to have a connected planar graph with an odd number of faces and every vertex having degree 6.
Step-by-step explanation:
(a) According to Euler's formula for planar graphs, a connected planar graph with V vertices, E edges, and F faces satisfies the equation V - E + F = 2. Therefore, for a graph with 7 vertices and 10 edges, we have V = 7 and E = 10. Substituting these values into the formula, we get 7 - 10 + F = 2. Solving for F, we find F = 5. Since F is not equal to 1, it is not possible to draw a connected graph with 7 vertices and 10 edges where no edges cross, creating faces.
(b) Similarly, for a graph with 10 vertices and 8 edges, we have V = 10 and E = 8. Plugging these values into Euler's formula, we get 10 - 8 + F = 2. Solving for F, we find F = 0. Since F is equal to 0, it is possible for a graph with 10 vertices and 8 edges to be a connected planar graph.
(c) In order to have a connected planar graph with an odd number of faces where every vertex has degree 6, we need to satisfy Euler's formula V - E + F = 2 and the Handshaking Lemma, which states that the sum of the degrees of all vertices in a graph is equal to twice the number of edges. Since every vertex has degree 6, the sum of the degrees is 6V. So, for an odd number of faces, the number of faces F must be odd. However, if F is odd, then V - E + F = 2 - E + F is odd too, which means E must be even. But for every edge, there are two endpoints, so the sum of degrees would be twice the number of edges, which means the number of edges E must be even. Therefore, it is not possible to have a connected planar graph with an odd number of faces where every vertex has degree 6.