Also, do the following problems: 1. Prove the 7-coloring theorem for a torus as follows: (a) Use Euler's Formula for the torus to show that any graph diagram (with no crossing edges) on the torus has a vertex of order 6 or less (modify the proof of the Planar Graph Fact). (b) Modify the 6-coloring method of Chapter 1 to show that any graph diagram on the torus can be 7-colored. 2. Give an example of a graph diagram on the torus that requires 7 colors. (Hint: you already drew this diagram in an earlier exercise in this assignment.) Also, do the following problems: 1. Prove the 7-coloring theorem for a torus as follows: (a) Use Euler's Formula for the torus to show that any graph diagram (with no crossing edges) on the torus has a vertex of order 6 or less (modify the proof of the Planar Graph Fact). (b) Modify the 6-coloring method of Chapter 1 to show that any graph diagram on the torus can be 7-colored. 2. Give an example of a graph diagram on the torus that requires 7 colors. (Hint: you already drew this diagram in an earlier exercise in this assignment.)