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The first non-obvious case of Hadwiger's Conjecture states that every graph$G$ with chromatic number at least 4 has a $K_4$ minor. We give a new proofthat derives the $K_4$ minor from a proper 3-coloring of a subgraph of $G$.

A note on Hadwiger's conjecture: Another proof that every 4-chromatic graph has a $K_4$ minor