Prove that every planar graph has at least one vertex of degree at most five. Conclude that planar graphs are six-colorable.
Prove that every triangle-free planar graph has at least one vertex of degree at most three. Conclude that such graphs are four-colorable.
Notes : Planar graphs are known to be 4-colorable (Four Color Theorem). Triangle-free planar graphs are known be three-colorable (Grotzsch's theorem).
Posted: Jun 29 '12
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Last updated: Oct 19 '12