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

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 $r,s$.

On the crossing numbers of join products of five graphs of order six with the discrete graph

“…The methods presented in the paper are based on combinatorial properties of cyclic permutations. Some of the ideas and methods were used for the first time in [3,11]. In [2,12,13], the properties of cyclic permutations are verified with the help of the software described in [1].…”

“…Let D be a good drawing of the graph G * + D n . The rotation rot D (t i ) of a vertex t i in the drawing D is the cyclic permutation that records the (cyclic) counterclockwise order in which the edges leave t i , as defined by Hernández-Vélez et al [3]. We use the notation (123456) if the counterclockwise order of the edges incident with the vertex 6 .…”

“…For more details, see the considered subdrawing of G * ∪ T k in Figure 3. 3 , and v 6 of G * on its boundary. This forces that no edge of the graph G * ∪ T k is crossed by an edge t l v j , for j = 1, 2, 3, 6.…”

Determining Crossing Number of Join of the Discrete Graph with Two Symmetric Graphs of Order Five

“…Let D be a drawing of the graph G + D n . The rotation rot D (t i ) of a vertex t i in the drawing D like the cyclic permutation that records the (cyclic) counter-clockwise order in which the edges leave t i has been defined by Hernández-Vélez, Medina, and Salazar [13]. We use the notation (12345) if the counter-clockwise order the edges incident with the vertex t i is…”

“…It turns out that if the graph of configurations is used like a graphical representation of the minimum numbers of crossings between two different subgraphs, then the proof of the main theorem will be simpler to understand. Similar methods were partially used for the first time in the papers [8,13]. In [4,9,10,14], the properties of cyclic permutations were also verified with the help of software in [15].…”