Question: Let $G$ be a connected graph with 3 or more vertices and such that $chi(G)=3$. Consider a proper 3-coloring of $G$ with colors, red, blue,

Let $G$ be a connected graph with 3 or more vertices and such that $\chi(G)=3$. Consider a proper 3-coloring of $G$ with colors, red, blue, and green. Prove that there exists a green node that has both a red neighbor and a blue neighbor.

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