Shellable Drawings and the Cylindrical Crossing Number of $$K_{n}$$ K n

Abstract

The Harary-Hill Conjecture states that the number of crossings in any drawing of the complete graph K n in the plane is at least Z (n) := 1

Levi's Lemma, pseudolinear drawings of , and empty triangles

“…Apply (3.2) to each of the 4 's induced by , , 2 , 1 and , , 2 , 1 . The conclusion is that 2 is the crossing of 1 1 with 2 2 . However, both 2 and 2 are in 1 , showing that 2 2 must cross 1 an even number of times.…”

“…The Harary–Hill Conjecture asserts that the crossing number of the complete graph is equal to The work of Ábrego et al. verifies this conjecture for “shellable” drawings of ; this is one of the first works that identifies a topological, as opposed to geometric, criterion for a drawing to have at least crossings.…”

A note on the cross-index of a complete graph based on a linear tree

“…Example 2.4. For n = 14, σ(a (1,7) ) is obtained as follows (see Figure 7): σ(a (1,7) ) = The components a (k,l) of M n satisfying 1 < k < 7 < l and a (k,l) = a (1,7) = 1 are a (2,8) , a (2,9) , a (2,10) , a (2,11) , a (2,12) , a (3,8) , a (3,9) , a (3,10) , a (3,11) , a (4,8) , a (4,9) , a (4,10) , a (5,8) , a (5,9) and a (6,8) .…”

“…It is unknown if there exists such a based diagram for any even number n ≥ 10. In [2], a condition "shellable" for a diagram D of K n is introduced, and it is proved that if D is shellable then c(D) ≥ Z(n). In [2] it is also proved that any based diagram of (K n ; H) is shellable.…”

“…σ(a (2,10) The components a (k,l) of M n satisfying 2 < k < 10 < l and a (k,l) = a (2,10) = 1 are a (3,11) , a (8,14) , a (9,13) and a (9,14) .…”