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Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.

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Does every graph admit an embedding such that identically-colored edges do not cross?

Given a graph, is it always possible to color the edges of the graph using two colors such that there exists an embedding of the graph in the plane where only opposite-colored edges cross? Simple coun …