A simple graph that is isomorphic to its complement is self-complementary. (i) Prove that, if G is self-complementary, then G has 4k or 4k+1 vertices, where k is an integer. (ii) Find all self-complementary graphs with 4 and 5 vertices. (iii) Find a self-complementary graph with 8 vertices.

Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a common vertex.

2) Parts (a) and (b) on this page are separate. (a) An undirected simple graph has 1000 vertices of degree 6 and 200 vertices of de- gree 2. This accounts for all vertices. How many edges does it have? Work this out to a final numeric answer. (b) How many directed simple graphs (no loops or multiple edges) are there on vertex set {0, 1,...,9}? Express your answer to (b) using an appropriate formula with specific numbers plugged in, such as V100! + 2, rather than evaluating it to a final numerical result.

A graph is bipartite if its vertex set can be partitioned into two sets V₁ and V2 such all edges are between V₁ and V2 (i.e. there are no edges joining vertices inside V₁, and the same for V2). (a) Draw a bipartite graph with 5 vertices and 5 edges. (b) What is the maximum number of edges for a bipartite graph with 2n vertices (suppose n > 1)?