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Prove or disprove the following: (a) Every 3-regular planar graph has a 3-coloring. (b) If ?=(?,?)...

Prove or disprove the following:

(a) Every 3-regular planar graph has a 3-coloring.

(b) If ?=(?,?) is a 3-regular graph and there exists a perfect matching of ?, then there exists a set of edges A⊆E such that each component of G′=(V,A) is a cycle

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Answer #1

(a) False.

Take G=K​​​​​​4, the complete graph with 4 vertices. This is a 3-regular graph with chromatic number 4. K​​​​​​4has no three coloring.

(b) True.

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