Answer to Question #139447 in Discrete Mathematics for Nanditha

Question

. The crossing number of a graph G, written ν(G), is the fewest number of nonendpoint edge-crossings that occur over all possible drawings of G in the plane. We assume that no edge crosses itself, and that edge crossings occur only at pairs of edges (ie, no three edges can cross at one point). With these conventions, we can say that G is planar if and only if ν(G) = 0.
(a) Prove that ν(K5) = 1.
(b) Prove that ν(K6) = 3 by
• exhibiting a drawing of K6 with exactly three edge-crossings, and
• assuming that K6 can be drawn with two edge crossings, introducing new vertices at the two edgecrossings, and then using Euler’s Formula to try to obtain a contradiction.

Answer 1

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