Share
Explore BrainMass

Prove that every regular tournament is strong.

Prove that every regular tournament is strong.

Hints
Here we need to first figure out something more about outdegrees and indegrees and orders of regular digraphs. Try to find a regular digraps with 3, 4, 5 ,6 vertices, and generalize.

D is a regular tournament if there is k such that outdegree x = k and
indegree x = n-k-1 for every x in D.

Please can you explain what does regular tournament strong mean?
Can you give a graph or graphs.

Explain this problem step by step.

Solution Preview

Let's study what a regular tournament looks like.

D is a regular tournament, from definition, we can find some k, such that
for each vertex x in D, x has indegree k and outdegree n-k-1. We have the
following inductions from this definition.

(1) We consider an edge u->v. This edge gives vertex u an indegree, but
also gives vertex v an outdegree. So in a directed graph, the total number
of indegrees is always equal to the totol number of outdegrees. Since all
the vertices in D have the same indegrees and outdegrees, we must ...

Solution Summary

It is proven that every regular tournament is strong. The solution is detailed and well presented.

$2.19