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Say we have a graph with 6 nodes/vertices. Say that within this graph, we draw 12 edges connecting the nodes. Why can we guarantee that at least one of the nodes will have 3 edges connected to it?

I understand why this has to be true when drawn out on paper, but I am wondering if there is a principle or theory that could help in explaining this. I'd like to understand it from a mathematical perspective rather than pure intuition. Thanks!

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If all $6$ nodes have degree $3$ or less, then there will be at most $6\cdot 3=18$ ends of edges, which means at most $\frac{18}{2}=9$ edges in total. This contradicts your specification that there must be $12$ edges.

So in fact there must be at least one node with degree $4$ or more.

(You cannot guarantee that there must be a node of degree exactly $3$: the octahedral graph a 4-regular graph with 6 nodes and 12 edges).

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