# Proof in Graph Theory

I have a $2D$ undirected graph of size $n \times n$ in which each node is connected to its four neighbours (left,right,top,bottom).

If any general property is true for any nxn graph, what will be the mathematical proof for the property to hold for $(n+1) \times (n+1)$ undirected graph? All nodes are logically equivalent i.e. have some algorithm running on them.

Is induction a good choice, if so, can someone kindly give hints if the base case is $3 \times 3$.

Thanking you in advance.

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The proof depends on the property. And if all have four neighbours - is the graph rolled to a torus? – Hagen von Eitzen Apr 8 '14 at 10:08
Thank you very much for your kind response. Graph is not rolled to a torus i.e. The edge nodes have only 2 or 3 neighbours. In fact, it is a square grid graph. – Curious Apr 9 '14 at 6:39
Property is any general property. is it not possible to prove it in general, without specifying any property? – Curious Apr 9 '14 at 6:41

## 1 Answer

What you have defined is a grid graph. You may find the properties of a grid graph in following links:

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Thank you very much for the technical correction. Yes, exactly this is a square grid graph. – Curious Apr 9 '14 at 6:41
Can you please suggest any hint on how to prove or disprove the argument? – Curious Apr 10 '14 at 7:43
If you're writing a paper, you can refer to the articles. Else, you can read the papers and copy the proofs from the papers. – cagirici Apr 10 '14 at 19:12