Problem 3 (5 points each). Draw a graph with the specified properties or show that no such graph exists. a. A graph with 4 vertices of degrees 1, 1, 2, and 3. b. A graph with 4 vertices of degrees 1, 1, 3, and 3. c. A simple graph with 4 vertices of degrees 1, 1, 3, and 3. d. A simple graph contains 8 vertices with degrees 0, 1, 2, 3, 4, 5, 6, 7. e. A simple graph contains 4 vertices and 12 edges. f. A graph contains 4 vertices with degrees 1, 2, 2, 3.

Problem 4 (15 points) A simple graph is called 3-regular if every vertex has degree 3. Show that any 3-regular graph has an even number of vertices. (Hint: use the Handshaking Theorem).