On the Crossing Number of Complete Graphs with an Uncrossed Hamiltonian Cycle

Guy proved his conjecture for all n ≤ 10. This was later extended by Pan and Richter in [3] to n ≤ 12. For general n, the most that is known is that 0.8594Z(n) ≤ cr(Kn) ≤ Z(n). The lower bound was proved by de Klerk et al. in [2]. The upper bound can be obtained in a number of ways, one of which we present here. Draw the Kn on the double cover of a disc, glued along the boundary, which is of course isomorphic to the sphere. Place the n vertices around the boundary of the circle, labeled clockwise by elements of Z/n. Let S1 = {1, 2, . . . , bn/2c}, S2 = {bn/2c + 1, . . . , n}. For two vertices x, y connect them by the line segment in the top disc if x + y ∈ S1 and by the segment in the bottom disc in x+ y ∈ S2. For generic locations of the vertices, no three edges intersect at a point, and we claim that the number of crossings is given by Z(n). Note that a pair of edges (u, v) and (t, w) cross if and only if the corresponding cords of the circle cross and if u+ v and t+w are either both in S1 or both in S2. Letting the four points t, u, v, w when listed in clockwise order be given by the labels x, x + a, x + a + b, x + a + b + c for integers 1 ≤ a, b, c ≤ n with a+b+c < n. We note that of the three pairs of edges we could divide these four points into, only {(x, x+ a+ b), (x+ a, x+ a+ b+ c)} has a chance of crossing. Note also that each pair of edges can be represented like this in exactly four ways by picking each possible vertex as x. We note that unless a+ c = n/2 (in which case it is easy to verify that the two edges are drawn on opposite discs and thus don’t cross) that exactly 2 of these representations have a+ c < n/2. For such values of a, b, c, we wish to know the number of x that give crossing edge pairs. The above edges cross if and only if 2x+a+b and (2x+a+b)+(a+c) either both lie in S1 or both lie in S2. Consider the average of the number of such x, and the number of such x obtained when the roles of a and c are reversed. We claim that this is the number of y ∈ Z/n for which y and y + (a+ c) either