(a) For each of the degree sequences below, construct a simple graph with the given degree sequence, or else prove that a simple graph with the degree sequence is not possible. (i) Degree sequence: (1, 2, 3, 3, 4, 5, 6) (ii) Degree sequence: (1, 2, 3, 4, 5, 6, 7)

Let the degree sequence of a graph G be the sequence of length |V(G)| that contains the degrees of the vertices of G in non-increasing order. (a) For each of the following sequences, either draw a simple graph whose de- gree sequence is equal to that sequence, or explain why such a graph does not exist: (i) (4, 4, 4, 2, 2), (ii) (4, 2, 2, 1, 1), (iii) (3,3,3,2,1), (iv) (4, 3, 3, 2, 1), (v) (2, 2, 2, 1, 1).