Introduction to graph theory/Problems 2

Planar Graphs
The n-dimensional Cube $$Q_n$$ or hypercube can be seen as a graph. Its nodes are the words of length $$n$$ over the alphabet $$\{ 0,1 \}$$, or $$V(Q_n) = \{ 0,1 \}^n$$. Two nodes are adjacent if and only if they differ in exactly one position.


 * 1) How many nodes does $$Q_n$$ have? How many edges?
 * 2) Describe the degrees of the nodes in $$Q_n$$.
 * 3) Embed $$Q_1, Q_2, Q_3$$ and $$Q_4$$ into the plane - without intersections, if possible.
 * 4) Embed $$Q_4$$ intersection-free on the surface of a torus.

Skewness

The skewness of a graph is the minimum number of edges which have to be removed from $$G$$ so that the resulting graph is planar.
 * 1) Show that for a simple graph $$G$$ with $$n \ge 3$$ nodes and $$m$$ edges the following equation holds:
 * $$skewness(G) \ge m - 3n + 6$$


 * 1) Calculate the skewness for $$K_3, K_5, K_{3,3}$$

Trees

Show the equivalence of the following statements for a graph $$G$$ with $$n$$:
 * 1) $$G$$ is a tree, i.e. $$G$$ is connected and has circle-free.
 * 2) $$G$$ is connected and has $$n-1$$ edges.
 * 3) $$G$$ is circle-free and has $$n-1$$ edges.