Homework problems

Problem 1. There are 30 students in the class. Can it happen that nine of them have 3 friends each (in the class), eleven have 4 friends each, and ten have 5 friends each.

Problem 2. A group of islands are connected by bridges in such a way that one can walk from any island to any other. A tourist walked around every island, crossing each bridge exactly once. He visited the island Thrice three times. How many bridges are there to Thrice, if

(a) the tourist neither started nor ended on Thrice;

(b) the tourist started on Thrice, but didn't end there;

(c) the tourist started and ended on Thrice?

Problem 3. Prove that for planar graph $2E\geq3F$.

Hint. Every face is bounded by at least three edges.

Problem 4. Prove that in any planar graph there exists a vertex with degree no more than 5.

Hint. Use the previous problem.