300 Solutions to Exercises 10. Copy the proof of Theorem 10.1.4. The only change is that we have to be sure it is possible to follow an arc in the proper direction. 11. [BB] A digraph has an Eulerian trail between vertices u and v if and only if • one of these vertices, say u, has outdegree one more than its indegree, the other vertex, v, has indegree one more than its outdegree, • the indegrees and outdegrees of every vertex except u and v are equal, and • for every pair of vertices x, y there is a (directed) path from x to y or from y to x. Proof. If there is an Eulerian trail from u to v, it is clear that the three given conditions are necessary. On the other hand, suppose the three conditions hold in a digraph 9. We show that 9 has an Eulerian trail. The result is obvious if u and v are the only vertices of 9, so we may assume 9 has at least three vertices. x -_--.. .....T v By hypothesis, there is an arc ux. If x =f. v, there is an arc xy, and so on. By hypothesis, this process can be continued without repeating an arc. It
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Polyhedron, vertices, adjacency matrix, Eulerian Trail, arc yw