91Ó°ÊÓ

Give an example of a digraph that does not have a closed Eulerian directed trail but whose underlying general graph has a closed Eulerian trail.

Prove that a tournament is strongly connected if and only if it has a directed Hamilton cycle.

Prove that every tournament contains a vertex \(u\) such that, for every other vertex \(x\), there is a path from \(u\) to \(x\) of length at most \(2 .\)

Prove that every graph has the property that it is possible to orient each of its edges so that, for each vertex \(x\), the indegree and outdegree of \(x\) differ by a! most 1 .