The conjecture cr(C_m\times C_n)=(m-2)n is true for all but finitely many n, for each m

It has been long congectured that the crossing number of $C_m\times C_n$ is $(m-2)n$ for $2<m<=n$. In this paper we proved that conjecture is true for all but finitely many $n$ for each $m$. More specifically we proved conjecture for $n>=(m/2)((m+3)^2/2+1)$.The proof is largely based on the theory of arrangements introduced by Adamsson and further developed by Adamsson and Richter.Comment: 16 pages, plainTeX, to be subnitted to "J. of Graph Theory

Large area convex holes in random point sets

Let $K, L$ be convex sets in the plane. For normalization purposes, suppose that the area of $K$ is $1$. Suppose that a set $K_n$ of $n$ points are chosen independently and uniformly over $K$, and call a subset of $K$ a {\em hole} if it does not contain any point in $K_n$. It is shown that w.h.p. the largest area of a hole homothetic to $L$ is $(1+o(1)) \log{n}/n$. We also consider the problems of estimating the largest area convex hole, and the largest area of a convex polygonal hole with vertices in $K_n$. For these two problems we show that the answer is $\Theta\bigl(\log{n}/n\bigr)$

The knots that lie above all shadows

We show that for each even integer $m\ge 2$, every reduced shadow with sufficiently many crossings is a shadow of a torus knot T(2,m+1), or of a twist knot $T_m$, or of a connected sum of $m$ trefoil knots

Closing in on Hill's conjecture

Borrowing L\'aszl\'o Sz\'ekely's lively expression, we show that Hill's conjecture is "asymptotically at least 98.5% true". This long-standing conjecture states that the crossing number cr($K_n$) of the complete graph $K_n$ is $H(n) := \frac{1}{4}\lfloor \frac{n}{2}\rfloor \lfloor \frac{n-1}{2}\rfloor \lfloor \frac{n-2}{2}\rfloor \lfloor\frac{n-3}{2}\rfloor$, for all $n\ge 3$. This has been verified only for $n\le 12$. Using flag algebras, Norin and Zwols obtained the best known asymptotic lower bound for the crossing number of complete bipartite graphs, from which it follows that for every sufficiently large $n$, cr$(K_n) > 0.905\, H(n)$. Also using flag algebras, we prove that asymptotically cr$(K_n)$ is at least $0.985\, H(n)$. We also show that the spherical geodesic crossing number of $K_n$ is asymptotically at least $0.996\, H(n)$.Comment: 20 pages, 5 figures, fixed remarks from referee

Balanced lines in two-coloured point sets

Let $B$ and $R$ be point sets (of {\em blue} and {\em red} points, respectively) in the plane, such that $P:=B\cup R$ is in general position, and $|P|$ is even. A line $\ell$ is {\em balanced} if it spans one blue and one red point, and on each open halfplane of $\ell$, the number of blue points minus the number of red points is the same. We prove that $P$ has at least $\min \{|B|,|R|\}$ balanced lines. This refines a result by Pach and Pinchasi, who proved this for the case $|B|=|R|$

On the decay of crossing numbers of sparse graphs

Richter and Thomassen proved that every graph has an edge $e$ such that the crossing number \ucr(G-e) of $G-e$ is at least (2/5)\ucr(G) - O(1). Fox and Cs. T\'oth proved that dense graphs have large sets of edges (proportional in the total number of edges) whose removal leaves a graph with crossing number proportional to the crossing number of the original graph; this result was later strenghtened by \v{C}ern\'{y}, Kyn\v{c}l and G. T\'oth. These results make our understanding of the {decay} of crossing numbers in dense graphs essentially complete. In this paper we prove a similar result for large sparse graphs in which the number of edges is not artificially inflated by operations such as edge subdivisions. We also discuss the connection between the decay of crossing numbers and expected crossing numbers, a concept recently introduced by Mohar and Tamon

Well-quasi-order of plane minors and an application to link diagrams

A plane graph $H$ is a {\em plane minor} of a plane graph $G$ if there is a sequence of vertex and edge deletions, and edge contractions performed on the plane, that takes $G$ to $H$. Motivated by knot theory problems, it has been asked if the plane minor relation is a well-quasi-order. We settle this in the affirmative. We also prove an additional application to knot theory. If $L$ is a link and $D$ is a link diagram, write $D\leadsto L$ if there is a sequence of crossing exchanges and smoothings that takes $D$ to a diagram of $L$. We show that, for each fixed link $L$, there is a polynomial-time algorithm that takes as input a link diagram $D$ and answers whether or not $D\leadsto L$

Toroidal grid minors and stretch in embedded graphs

We investigate the toroidal expanse of an embedded graph G, that is, the size of the largest toroidal grid contained in G as a minor. In the course of this work we introduce a new embedding density parameter, the stretch of an embedded graph G, and use it to bound the toroidal expanse from above and from below within a constant factor depending only on the genus and the maximum degree. We also show that these parameters are tightly related to the planar crossing number of G. As a consequence of our bounds, we derive an efficient constant factor approximation algorithm for the toroidal expanse and for the crossing number of a surface-embedded graph with bounded maximum degree

The optimal drawings of K_{5,n}

Zarankiewicz's Conjecture (ZC) states that the crossing number cr$(K_{m,n})$ equals Z(m,n):=\floor{\frac{m}{2}} \floor{\frac{m-1}{2}} \floor{\frac{n}{2}} \floor{\frac{n-1}{2}}. 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 $r,s$.Comment: In the previous version the bibliography was missing. Otherwise this is identical as Version

The unavoidable arrangements of pseudocircles

It is known that cyclic arrangements are the only {\em unavoidable} simple arrangements of pseudolines: for each fixed $m\ge 1$, every sufficiently large simple arrangement of pseudolines has a cyclic subarrangement of size $m$. In the same spirit, we show that there are three unavoidable arrangements of pseudocircles