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I was given this problem to solve by a professor who promised an A if anyone could solve it. I'm nearly certain it is impossible, because at some point you have too many vertices and inevitably box yourself in, but I would like to know how to prove it.

How can you prove that it is impossible to connect every * with every 0 without overlapping lines?

*    *    *   

0    0    0

EDIT: Professor said you get an A if you solve it, not prove it's impossible. I'm not even in the class, it was my friend's question, I am just curious.

OneChillDude
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  • Well, it looks like you will be getting a B then :) – MathMajor Feb 12 '15 at 06:49
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    So, if I tell you the answer, then do I get the A? – Blue Feb 12 '15 at 06:51
  • Your profs is too old school that he/she doesn't even know there 's something called math stackexchage? Half of the class will get A then. –  Feb 12 '15 at 06:54
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    The prof only offered an A for a solution, not necessarily for a proof that it's imposible! – mrf Feb 12 '15 at 06:56
  • There are two proof of this in Chapter 5 of Robin. J. Wilson's book "Introduction to graph theory". One is constructive, the other one uses Euler's formula. If you still are not satisfied, you can look at Chapter 4 of Richard Diestel's book "Graph Theory" (GTM 173) which uses topology to prove the same statement rigorously. – achille hui Feb 12 '15 at 07:31
  • The drawing requested is a well-known $K_{3,3}$ 'utility graph' http://en.wikipedia.org/wiki/Water,_gas,_and_electricity which is known to be not planar http://en.wikipedia.org/wiki/Planar_graph – so the task is impossible. – CiaPan Feb 12 '15 at 07:33

1 Answers1

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The problem of connecting these points using edges is a Graph-Theoretic Problem. If you are unfamiliar with graph theory, I suggest readng up on it here.

Specifically, we say that a graph is planar if it can be drawn such that no edges overlap, as you are trying to do. Wagner's theorem states that a graph is planar only if it does not contain $K_5$ or $K_{3,3}$ as a minor.

But the graph you're trying to achieve -- connecting every * with every 0 -- is $K_{3,3}$. And every graph is a minor of itself, so the graph you're trying to achieve contains $K_{3,3}$ as a minor, so it is not planar.

You'll probably want to read up on complete graphs to understand this answer. $K_{3,3}$ is the complete bipartite graph on two sets of three vertices.

Newb
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