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The girth of a graph G is the length of its shortest cycle (if G has no cycles, then its girth is defined to be infinite). (a) (6 points) Let G be a connected planar graph with at least 4 vertices and girth at least 6. Show that G has a vertex of degree at most 2. (b) (3 points) Recall that every planar graph is 4-colorable, and note that the previous part can be used to show that every planar graph with girth at least 6 is 3-colorable. Prove or disprove the following statement. "There exists a positive integer g such that every planar graph with girth at least g is 2-colorable."