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In other words, can a planar graph without k3 have a chromatic number larger than 3?

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up vote 4 down vote accepted

Yes, every triangle-free planar graph is 3-colorable. This is known as Grötzsch's theorem.

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It is perhaps worth noting that the theorem does not generalize in the natural way, as evidenced in answers to this question:… – Austin Mohr Nov 1 '12 at 2:58

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