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