0
$\begingroup$

Imagine a city that has 15 public phones. Is it possible to connect them to each other with some cables in case that every phone must connect to exactly 5 another phone. I tried to draw this graph with 15 vertices but

I could just fill 14 vertices and the last vertex 's degree was 4.

Is there a strong way to prove that it's possible?

$\endgroup$
2
$\begingroup$

Hint In any graph, the number of vertices of odd degree must be even.

$\endgroup$
3
  • $\begingroup$ Is there always a graph with $n$ nodes, such that each node is connected to $k<n$ other nodes (for any choices of $n$ and $k$ that meet the criterion you stated)? $\endgroup$ – DreiCleaner Jun 16 '20 at 15:13
  • $\begingroup$ @DreiCleaner It is possible if and only if $n \cdot k$ is even. $\endgroup$ – N. S. Jun 16 '20 at 15:56
  • 1
    $\begingroup$ @Integrand What do you mean by "It does not provide an answer"? 15 is ODD. $\endgroup$ – N. S. Jun 16 '20 at 15:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.