Abstract
Zarankiewicz's Conjecture (ZC) states that the crossing number cr$(K_{m,n})$ equals $Z(m,n):=\lfloor{\frac{m}{2}}\rfloor \lfloor{\frac{m-1}{2}}\rfloor \lfloor{\frac{n}{2}}\rfloor \lfloor{\frac{n-1}{2}}\rfloor$. Since Kleitman's verification of ZC for $K_{5,n}$ (from which ZC for $K_{6,n}$ 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 $K_{5,n}$. The widely known natural drawings of $K_{m,n}$ (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 $K_{m,2}$ has no crossings. Antipodal vertices also play a major role in Kleitman's inductive proof that cr$(K_{5,n}) = Z(5,n)$. We explore in depth the role of antipodal vertices in optimal drawings of $K_{5,n}$, for $n$ even. We prove that if {$n \equiv 2$ (mod $4$)}, then every optimal drawing of $K_{5,n}$ has antipodal vertices. We also exhibit a two-parameter family of optimal drawings $D_{r,s}$ of $K_{5,4(r+s)}$ (for $r,s\ge 0$), with no antipodal vertices, and show that if $n\equiv 0$ (mod $4$), then every optimal drawing of $K_{5,n}$ without antipodal vertices is (vertex rotation) isomorphic to $D_{r,s}$ for some integers $r,s$. As a corollary, we show that if $n$ is even, then every optimal drawing of $K_{5,n}$ is the superimposition of Zarankiewicz drawings with a drawing isomorphic to $D_{r,s}$ for some nonnegative integers $r,s$.
Highlights
We explore in depth the role of antipodal vertices in optimal drawings of K5,n, for n even
We prove that if n ≡ 2, every optimal drawing of K5,n has antipodal vertices
We show that if n is even, 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
Summary
We recall that the crossing number cr(G) of a graph G is the minimum number of pairwise crossings of edges in a drawing of G in the plane. Antipodal pairs are crucial in the inductive step of Kleitman’s proof, which does not concern itself with the different ways (if more than one) to achieve Z(5, n) crossings with a drawing of K5,n Given their preeminence in Zarankiewicz’s Conjecture, we set out to investigate the role of antipodal pairs in the optimal drawings of K5,n. This abstraction (and the related concept of core) will prove to be extremely useful for the electronic journal of combinatorics 21(4) (2014), #P4.1 the proof of Theorem 1. The proof of Theorem 1, given in Section 13, is an easy consequence of this full characterization of cores
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