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Suppose we are going to color the vertices of a connected planar simple graph such that...

Suppose we are going to color the vertices of a connected planar simple graph such that no two adjacent vertices are with the same color.

(a) Prove that if G is a connected planar simple graph, then G has a vertex of degree at most five.

(b) Prove that every connected planar simple graph can be colored using six or fewer colors.

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