Show that if a circuit in a planar graph encloses exactly two regions, each of which has an even number of boundary edges, then the circuit has even length

Complete a path straight at vertex a that ends at vertex c.

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)?