A directed graph is simple if each ordered pair of vertices is the head and tail of at most one edge; one loop may be present at each vertex. For each n ≥ 1, prove or disprove the following statement. Every simple directed graph with n vertices has two vertices with the same outdegree or two vertices with the same indegree.

(a) How many words of length 2 are there over the alphabet A = { 0; 1; 2; 3}?

(b) Construct the de Bruijn digraph D_4,2 and use this digraph to nd an appropriate de Bruijn sequence.

(c) For b = 4 and n = 2, how many de Bruijn sequences can be formed?

Suppose [S, T] and [X, Y ] are two minimal cuts in a network N. Prove that both [S [X U T ∩ Y ] and [S ∩ X, T U Y ] are also minimal cuts in N.