Consider a drawing G of a (not necessarily planar) graph G in the plane. Two edges of G cross if they meet at a point other than a vertex of G. Each such point is called a crossing of the two edges. The crossing number of G, denoted by cr(G), is the least number of crossings in a drawing of G in the plane. For example, cr(G) = 0 if and only if G is planar.

(a) Show that cr(K5) = cr(K3,3) = 1.

(b) Show that cr(X) = 2, where X is the Petersen graph (Hint 1: To show that cr(X) 2, you need to find a drawing with 2 crossings. Hint 2: To show cr(X) 2, suppose cr(X) = 1 (since it is known to be non-planar) and use the fact that the shortest cycle in X has length 5).