- 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.

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Nalin Pithwa