- An Eulerian trail in a digraph is a trail containing all the edges. An Eulerian circuit is a closed trail containing all the edges. Show that a digraph X contains an Eulerian circuit if and only if for every vertex v and the underlying graph has at most one component.
- Determine for what values of and is Eulerian.
- What is the maximum number of edges in a connected, bipartite graph of order n?
- How many 4-cycles are in ?
- Let be the n-dimensional cube graph. Its vertices are all the n-tuples of 0 and 1 with two vertices being adjacent if they differ in precisely one position. Show that is connected and bipartite.