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For example, how to prove that every realization of a graph of this degree sequence $\left(2,2,2,2,4,4,4,6\right)$ is connected?

Misha Lavrov
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abbief
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Jan 13 '23 at 19:51
  • The verb is "prove", not "proof". "Proof" is a noun. (Unless the graph has yeast in it.) – Misha Lavrov Jan 13 '23 at 19:56

1 Answers1

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The last vertex is connected to all but one. The remaining vertex has degree two or four, so it is connected to some other vertex. This completes the proof.

As for other examples, looking at minimal and maximal degrees is a good idea. Thinking about the number of vertices and the number of edges is also important. For example, here the handshaking lemma gives that there are thirteen edges.

If you want further information, here is an answer with extensive information.

Matija
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