The Optimal Drawings of $K_{5, n}$

Zarankiewicz’s Conjecture (ZC) states that the crossing number cr(Km,n) equals Z(m,n) := bm2 cb m−1 2 cb n 2 cb n−1 2 c. Since Kleitman’s verification of ZC for K5,n (from which ZC for K6,n easily follows), very little progress has been made around ZC; the most notable exceptions involve computer-aided results. With the aim of gaining a more profound understanding of this notoriously difficult conjecture, we investigate the optimal (that is, crossing-minimal) drawings of K5,n. The widely known natural drawings of Km,n (the so-called Zarankiewicz drawings) with Z(m,n) crossings contain antipodal vertices, that is, pairs of degree-m vertices such that their induced drawing of Km,2 has no crossings. Antipodal vertices also play a major role in Kleitman’s inductive proof that cr(K5,n) = Z(5, n). We explore in depth the role of antipodal vertices in optimal drawings of K5,n, for n even. We prove that if n ≡ 2 (mod 4), then every optimal drawing of K5,n has antipodal vertices. We also exhibit a two-parameter family of optimal drawings Dr,s of K5,4(r+s) (for r, s > 0), with no antipodal vertices, and show that if n ≡ 0 (mod 4), then every optimal drawing of K5,n without antipodal vertices is (vertex rotation) isomorphic to Dr,s for some integers r, s. As a corollary, we show that if n is even, then every optimal drawing of K5,n is the superimposition of Zarankiewicz drawings with a drawing isomorphic to Dr,s for some nonnegative integers r, s.