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A cluster in a drawing of a graph in the plane is the set of four endpoints of the two edges involved in a crossing. A independent drawing is a drawing in which the clusters are pairwise disjoint. Albertson [1] asked for the maximum k such that every independent drawing with k crossings has an independent set consisting of one vertex from each cluster. We prove that this maximum is 4. A drawing of a graph in the plane maps the vertices into points and the edges into continuous curves so that the endpoints of an edge become endpoints of the corresponding curve. A crossing is a common internal point on two edges. The four endpoints of the edges in a crossing form a cluster. A drawing is independent if its clusters are pairwise disjoint. A good set in a drawing is an independent set of vertices consisting of one vertex from each cluster. Albertson [1] asked whether every graph having an independent drawing is 5-colorable. He proved this for independent drawings with at most three crossings by finding a good set. When a drawing has a good set S, deleting a crossing edge incident to the vertex of S in each cluster yields a planar graph, and then the Four Color Theorem leads to a proper 5-coloring after replacing the missing matching. Albertson asked also for the maximum k such that every independent drawing with k crossings ha

Year: 2008

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