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Determining the exact number of non-planar graphs with 6 vertices is a complex combinatorial problem without a simple closed formula, but here are important points and known facts related to planarity and graphs with 6 vertices: *** ## Key Facts About Planar and Non-Planar Graphs on 6 Vertices - The total number of graphs on 6 labeled vertices is $$2^{\binom{6}{2}} = 2^{15} = 32768$$. - Many of these graphs are planar, but some are non-planar. - Classic non-planar graphs with 6 vertices include: - The complete graph $$K_5$$ (5 vertices, 10 edges) plus an isolated vertex is sometimes considered with 6 vertices. - The complete bipartite graph $$K_{3,3}$$, which has 6 vertices and 9 edges, is a classic example of a non-planar graph. - By _________ ___ _________ ____ ____ _________.
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