It has been long--conjectured that the crossing number cr(K_m,n) of the complete bipartite graph K_m,n equals the Zarankiewicz Number Z(m,n):= floor((m-1)/2) floor(m/2) floor((n-1)/2) floor(n/2). Another long--standing conjecture states that the crossing number cr(K_n) of the complete graph K_n equals Z(n):= floor(n/2) floor((n-1)/2) floor((n-2)/2) floor((n-3)/2)/4. In this paper we show the following improved bounds on the asymptotic ratios of these crossing numbers and their conjectured values:
(i) for each fixed m >= 9, lim_{n->infty} cr(K_m,n)/Z(m,n) >= 0.83m/(m-1);
(ii) lim_{n->infty} cr(K_n,n)/Z(n,n) >= 0.83; and
(iii) lim_{n->infty} cr(K_n)/Z(n) >= 0.83.
The previous best known lower bounds were 0.8m/(m-1), 0.8, and 0.8, respectively. These improved bounds are obtained as a consequence of the new bound cr(K_{7,n}) >= 2.1796n^2 - 4.5n. To obtain this improved lower bound for cr(K_{7,n}), we use some elementary topological facts on drawings of K_{2,7} to set up a quadratic program on 6! variables whose minimum p satisfies cr(K_{7,n}) >= (p/2)n^2 - 4.5n, and then use state--of--the--art quadratic optimization techniques combined with a bit of invariant theory of permutation groups to show that p >= 4.3593.
It has been long conjectured that the crossing numbers of the complete bipartite graph K_{m,n} and of the complete graph K_n equal Z(m,n) (the value conjectured by Zarankiewicz, who came up with a drawing reaching this value) and Z(n) :=Z(n,n-2)/4, respectively. In a 2-page drawing of a graph, the vertices are drawn on a straight line (the spine), and each edge is contained in one of the half-planes of the spine. The 2-page crossing number v_2(G) of a graph G is the minimum number of crossings in a 2-page drawing of G. Somewhat surprisingly, there are 2-page drawings of K_{m,n} (respectively, …
A "book with k pages" consists of a straight line (the "spine") and k half-planes (the "pages"), such that the boundary of each page is the spine. If a graph is drawn on a book with k pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is a k-page book drawing (or simply a k-page drawing). The k-page crossing number nu_k(G) of a graph G is the minimum number of crossings in a k-page drawing of G. In this paper we investigate the k-page crossing numbers of complete graphs K_n. We use semidefinite programming techniques to give improved lower …
In this note we provide an improved upper bound on the biplanar crossing number of the 8-dimensional hypercube. The $k$-planar crossing number of a graph $cr_k(G)$ is the number of crossings required when every edge of $G$ must be drawn in one of $k$ distinct planes. It was shown in Czabarka et al. that $cr_2(Q_8) \leq 256$ which we improve to $cr_2(Q_8) \leq 128$. Our approach highlights the relationship between symmetric drawings and the study of $k$-planar crossing numbers. We conclude with several open questions concerning this relationship.
The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a drawing of $G$. Motivated by the recent work [Faria, L., Figueiredo, C.M.H. de, Sykora, O., Vrt'o, I.: An improved upper bound on the crossing number of the hypercube. J. Graph Theory {\bf 59}, 145--161 (2008)] which solves the upper bound conjecture on the crossing number of $n$-dimensional hypercube proposed by Erdős and Guy, we give upper and lower bounds of the crossing number of locally twisted cube, which is one of variants of hypercube.
The crossing number of a graph $G$, ${\mbox{cr}}(G)$, is the minimum number of crossings, the pair-crossing number, ${\mbox{pcr}}(G)$, is the minimum number of pairs of crossing edges over all drawings of $G$. In this note we show that ${\mbox{cr}}(G)=O({\mbox{pcr}}(G)^{3/2}\log{\mbox{pcr}}(G))$, which is an improvement of the result of Matoušek, by a log factor.
The crossing number of a graph G is the minimum number of pairwise intersections of edges in a drawing of G. Motivated by the recent work [Faria, L., Figueiredo, C.M.H. de, Sykora, O., Vrt'o, I.: An improved upper bound on the crossing number of the hypercube. J. Graph Theory 59, 145-161 (2008)], we give an upper bound of the crossing number of n-dimensional bubble-sort graph Bn.
A graph $G$ is {$k$-crossing-critical} if $cr(G)\ge k$, but $cr(G\setminus e)<k$ for each edge $e\in E(G)$, where $cr(G)$ is the crossing number of $G$. It is known that for any $k$-crossing-critical graph $G$, $cr(G)\le 2.5k+16$ holds, and in particular, if $δ(G)\ge 4$, then $cr(G)\le 2k+35$ holds, where $δ(G)$ is the minimum degree of $G$. In this paper, we improve these upper bounds to $2.5k +2.5$ and $2k+8$ respectively. In particular, for any $k$-crossing-critical graph $G$ with $n$ vertices, if $δ(G)\ge 5$, then $cr(G)\le 2k-\sqrt k/2n+35/6$ holds.
Sharir and Welzl [1] derived a bound on crossing-free matchings primarily based on solving a recurrence based on the size of the matchings. We show that the recurrence given in Lemma 2.3 in Sharir and Welzl can be improve to $(2n-6s)\textbf{Ma}_{m}(P)\leq\frac{68}{3}(s+2)\textbf{Ma}_{m-1}(P)$ and $(3n-7s)\textbf{Ma}_{m}(P)\leq44.5(s+2)\textbf{Ma}_{m-1}(P)$, thereby improving the upper bound for crossing-free matchings.
This paper is a contribution to the problem of counting geometric graphs on point sets. More concretely, we look at the maximum numbers of non-crossing spanning trees and forests. We show that the so-called double chain point configuration of N points has Omega(12.52^N) non-crossing spanning trees and Omega(13.61^N) non-crossing forests. This improves the previous lower bounds on the maximum number of non-crossing spanning trees and of non-crossing forests among all sets of N points in general position given by Dumitrescu, Schulz, Sheffer and Tóth. Our analysis relies on the tools of analytic …
We define the crossing number for an embedding of a graph G into R^3, and prove a lower bound on it which almost implies the classical crossing lemma. We also give sharp bounds on the space crossing numbers of pseudo-random graphs.
We provide a new lower bound on the number of $(\leq k)$-edges of a set of $n$ points in the plane in general position. We show that for $0 \leq k \leq\lfloor\frac{n-2}{2}\rfloor$ the number of $(\leq k)$-edges is at least $$ E_k(S) \geq 3\binom{k+2}{2} + \sum_{j=\lfloor\frac{n}{3}\rfloor}^k (3j-n+3), $$ which, for $k\geq \lfloor\tfrac{n}{3}\rfloor$, improves the previous best lower bound in [J. Balogh, G. Salazar, Improved bounds for the number of ($\leq k$)-sets, convex quadrilaterals, and the rectilinear crossing number of $K_n$]. As a main consequence, we obtain a new lower bound on the re…
We present, to the best of the authors' knowledge, all known results for the (planar) crossing numbers of specific graphs and graph families. The results are separated into various categories; specifically, results for general graph families, results for graphs arising from various graph products, and results for recursive graph constructions.
The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. Our main result is that every graph $G$ that does not contain a fixed graph as a minor has crossing number $O(Δn)$, where $G$ has $n$ vertices and maximum degree $Δ$. This dependence on $n$ and $Δ$ is best possible. This result answers an open question of Wood and Telle [New York J. Mathematics, 2007], who proved the best previous bound of $O(Δ^2 n)$. We also study the convex and rectilinear crossing numbers, and prove an $O(Δn)$ bound for the convex crossing number of bounded pathwidth gr…
It is proved that the rectilinear crossing number of every graph with bounded tree-width and bounded degree is linear in the number of vertices. **** This paper has been withdrawn by the author. **** The results have been superseeded by the author's paper with Jan Arne Telle: "Planar decompositions and the crossing number of graphs with an excluded minor", http://arxiv.org/math/0604467.
Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags. The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. We prove that planar decompositions are intimately related to the crossing number. In particular, a graph with bounded degree has l…
The biplanar crossing number of a graph $G$ is the minimum number of crossings over all possible drawings of the edges of $G$ in two disjoint planes. We present new bounds on the biplanar crossing number of complete graphs and complete bipartite graphs. In particular, we prove that the biplanar crossing number of complete bipartite graphs can be approximated to within a factor of $3$, improving over the best previously known approximation factor of $4.03$. For complete graphs, we provide a new approximation factor of $3.17$, improving over the best previous factor of $4.34$. We provide similar…
A {\it good drawing} of a graph $G$ is a drawing where the edges are non-self-intersecting and each two edges have at most one point in common, which is either a common end vertex or a crossing. The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a good drawing of $G$ in the plane. The {\it $n$-dimensional augmented cube} $AQ_n$, proposed by S.A. Choudum and V. Sunitha, is an important interconnection network with good topological properties and applications. In this paper, we obtain an upper bound on the crossing number of $AQ_n$ less than $26/…
We investigate crossing minimization for 1-page and 2-page book drawings. We show that computing the 1-page crossing number is fixed-parameter tractable with respect to the number of crossings, that testing 2-page planarity is fixed-parameter tractable with respect to treewidth, and that computing the 2-page crossing number is fixed-parameter tractable with respect to the sum of the number of crossings and the treewidth of the input graph. We prove these results via Courcelle's theorem on the fixed-parameter tractability of properties expressible in monadic second order logic for graphs of bou…
We define a new kind of crossing number which generalizes both the bipartite crossing number and the outerplanar crossing number. We calculate exact values of this crossing number for many complete bipartite graphs and also give a lower bound.
We prove new lower bounds on the crossing number of a complete graphs assuming that it is drawn in such a way that it contains a Hamiltonian cycle with no crossings.
In this paper, we consider the embedding of a complete $d$-uniform geometric hypergraph with $n$ vertices in general position in $\mathbb{R}^d$, where each hyperedge is represented as a $(d-1)$-simplex, and a pair of hyperedges is defined to cross if they are vertex-disjoint and contains a common point in the relative interior of the simplices corresponding to them. As a corollary of the Van Kampen-Flores Theorem, it can be seen that such a hypergraph contains $Ω(\frac{2^d}{\sqrt{d}})$ $n\choose 2d$ crossing pairs of hyperedges. Using Gale Transform and Ham Sandwich Theorem, we improve this lo…
We prove crossing number inequalities for geometric graphs whose vertex sets are taken from a d-dimensional grid of volume N and give applications of these inequalities to counting the number of non-crossing geometric graphs that can be drawn on such grids. In particular, we show that any geometric graph with m >= 8N edges and with vertices on a 3D integer grid of volume N, has Ω((m^2/n)\log(m/n)) crossings. In d-dimensions, with d >= 4, this bound becomes Ω(m^2/n). We provide matching upper bounds for all d. Finally, for d >= 4 the upper bound implies that the maximum number of crossing-free …
The crossing number ${\mbox {cr}}(G)$ of a graph $G=(V,E)$ is the smallest number of edge crossings over all drawings of $G$ in the plane. For any $k\ge 1$, the $k$-planar crossing number of $G$, ${\mbox {cr}}_k(G)$, is defined as the minimum of ${\mbox {cr}}(G_1)+{\mbox {cr}}(G_2)+\ldots+{\mbox {cr}}(G_{k})$ over all graphs $G_1, G_2,\ldots, G_{k}$ with $\cup_{i=1}^{k}G_i=G$. Pach et al. [\emph{Computational Geometry: Theory and Applications} {\bf 68} 2--6, (2018)] showed that for every $k\ge 1$, we have ${\mbox {cr}}_k(G)\le \left(\frac{2}{k^2}-\frac1{k^3}\right){\mbox {cr}}(G)$ and that thi…
We prove that the number of unit distances among $n$ planar points is at most $1.94\cdot n^{4/3}$, improving on the previous best bound of $8n^{4/3}$. We also give better upper and lower bounds for several small values of $n$. We also prove some variants of the crossing lemma and improve some constant factors.
We prove that if $G$ is a graph with an minimal edge cut $F$ of size three and $G_1$, $G_2$ are the two (augmented) components of $G-F$, then the crossing number of $G$ is equal to the sum of crossing numbers of $G_1$ and $G_2$. Combining with known results, this implies that crossing number is additive over edge-cuts of size $d$ for $d\in\{0, 1, 2, 3\}$, whereas there are counterexamples for every $d\ge 4$. The techniques generalize to show that minor crossing number is additive over edge cuts of arbitrary size, as well as to provide bounds for crossing number additivity in arbitrary surfaces…
We verify an upper bound of Pach and Tóth [Combinatorica 17(1997), 427-439, Discrete and Computational Geometry 36(2006), 527-552] on the midrange crossing constant. Details of their $\frac{8}{9π^2}$ upper bound have not been available. Our verification is different from their method and hinges on a result of Moon [J. Soc. Indust. Appl. Math. 13(1965), 506-510]. As Moon's result is optimal, we raise the question whether the midrange crossing constant is $\frac{8}{9π^2}$.
A diagonal surface in a link exterior M is a properly embedded, incompressible, boundary incompressible surface which furthermore has the same number of boundary components and same slope on each component of the boundary of M. We derive a formula for the boundary slope of a diagonal surface in the exterior of a 2-bridge link which is analogous to the formula for the boundary slope of a 2-bridge knot found by Hatcher and Thurston. Using this formula we show that the diameter of a 2-bridge link, that is, the difference between the smallest and largest finite slopes of diagonal surfaces, is equa…
After $2$-crossing-critical graphs were characterized in 2016, their most general subfamily, large $3$-connected $2$-crossing-critical graphs, has attracted separate attention. This paper presents sharp upper and lower bounds for their domination and independence number.
Let $D$ be a straight-line drawing of a graph. The rectilinear 2-colored crossing number of $D$ is the minimum number of crossings between edges of the same color, taken over all possible 2-colorings of the edges of $D$. First, we show lower and upper bounds on the rectilinear 2-colored crossing number for the complete graph $K_n$. To obtain this result, we prove that asymptotic bounds can be derived from optimal and near-optimal instances with few vertices. We obtain such instances using a combination of heuristics and integer programming. Second, for any fixed drawing of $K_n$, we improve th…
The crossing number of a graph $G$ is the minimum number of edge crossings over all drawings of $G$ in the plane. A graph $G$ is $k$-crossing-critical if its crossing number is at least $k$, but if we remove any edge of $G$, its crossing number drops below $k$. There are examples of $k$-crossing-critical graphs that do not have drawings with exactly $k$ crossings. Richter and Thomassen proved in 1993 that if $G$ is $k$-crossing-critical, then its crossing number is at most $2.5k+16$. We improve this bound to $2k+6\sqrt{k}+44$.
Let $G$ be a multigraph with $n$ vertices and $e>4n$ edges, drawn in the plane such that any two parallel edges form a simple closed curve with at least one vertex in its interior and at least one vertex in its exterior. Pach and Tóth (A Crossing Lemma for Multigraphs, SoCG 2018) extended the Crossing Lemma of Ajtai et al. (Crossing-free subgraphs, North-Holland Mathematics Studies, 1982) and Leighton (Complexity issues in VLSI, Foundations of computing series, 1983) by showing that if no two adjacent edges cross and every pair of nonadjacent edges cross at most once, then the number of edge c…
In this paper, we consider a generalized version of the rectilinear crossing number problem of drawing complete graphs on a plane. The minimum number of crossing pairs of hyperedges in the $d$-dimensional rectilinear drawing of a $d$-uniform hypergraph is known as the $d$-dimensional rectilinear crossing number of the hypergraph. The currently best-known lower bound on the $d$-dimensional rectilinear crossing number of a complete $d$-uniform hypergraph with $n$ vertices in general position in $\mathbb{R}^d$ is $Ω(\frac{2^d}{\sqrt{d}} \log d) {n \choose 2d}$. In this paper, we improve this lowe…
Albertson conjectured that if graph $G$ has chromatic number $r$, then the crossing number of $G$ is at least that of the complete graph $K_r$. This conjecture in the case $r=5$ is equivalent to the four color theorem. It was verified for $r=6$ by Oporowski and Zhao. In this paper, we prove the conjecture for $7 \leq r \leq 12$ using results of Dirac; Gallai; and Kostochka and Stiebitz that give lower bounds on the number of edges in critical graphs, together with lower bounds by Pach et.al. on the crossing number of graphs in terms of the number of edges and vertices.
The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a drawing of $G$. The {\it $n$-dimensional folded hypercube} $FQ_n$ is a graph obtained from $n$-dimensional hypercube by adding all complementary edges. In this paper, we obtain upper and lower bounds of the crossing number of $FQ_n$.
We generalize the notion of non-crossing partition on a disk to general surfaces with boundary. For this, we consider a surface $Σ$ and introduce the number $C_Σ(n)$ of non-crossing partitions of a set of $n$ points laying on the boundary of $Σ$. Our proofs use bijective techniques arising from map enumeration, joint with the symbolic method and singularity analysis on generating functions. An outcome of our results is that the exponential growth of $C_Σ(n)$ is the same as the one of the $n$-th Catalan number, i.e., does not change when we move from the case where $Σ$ is a disk to general surf…
A graph drawn in the plane with n vertices is k-fan-crossing free for k > 1 if there are no k+1 edges $g,e_1,...e_k$, such that $e_1,e_2,...e_k$ have a common endpoint and $g$ crosses all $e_i$. We prove a tight bound of 4n-8 on the maximum number of edges of a 2-fan-crossing free graph, and a tight 4n-9 bound for a straight-edge drawing. For k > 2, we prove an upper bound of 3(k-1)(n-2) edges. We also discuss generalizations to monotone graph properties.
Given a surface with boundary and some points on its boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on a surface with any genus and number of boundary components. If only bigons are allowed, then it becomes an arc diagram. The count of arc diagrams is known to have a rich structure. We show that the count of polygon diagrams exhibits the same interesting behaviours, in particular it is almost polynomial in the number of points on the boundary components, and the leadi…
Scheinerman and Wilf (1994) assert that `an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph K_n.' A rectilinear drawing of K_n is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear, and that no three edges intersect in a point unless that point is an endpoint of all three. The rectilinear crossing number of K_n is the fewest number of edge crossings attainable over all rectilinear drawings of K_n. For each n we …
We consider spanning trees of $n$ points in convex position whose edges are pairwise non-crossing. Applying a flip to such a tree consists in adding an edge and removing another so that the result is still a non-crossing spanning tree. Given two trees, we investigate the minimum number of flips required to transform one into the other. The naive $2n-Ω(1)$ upper bound stood for 25 years until a recent breakthrough from Aichholzer et al. yielding a $2n-Ω(\log n)$ bound. We improve their result with a $2n-Ω(\sqrt{n})$ upper bound, and we strengthen and shorten the proofs of several of their resul…
A star-simple drawing of a graph is a drawing in which adjacent edges do not cross. In contrast, there is no restriction on the number of crossings between two independent edges. When allowing empty lenses (a face in the arrangement induced by two edges that is bounded by a 2-cycle), two independent edges may cross arbitrarily many times in a star-simple drawing. We consider star-simple drawings of $K_n$ with no empty lens. In this setting we prove an upper bound of $3((n-4)!)$ on the maximum number of crossings between any pair of edges. It follows that the total number of crossings is finite…
We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an $n$-vertex graph embedded on a surface of genus $g$ with at most $k$ crossings per edge has treewidth $O(\sqrt{(g+1)(k+1)n})$ and layered treewidth $O((g+1)k)$, and that these bounds are tight up to a constant factor. As a special case, the $k$-planar graphs with $n$ vertices have treewidth $O(\sqrt{(k+1)n})$ and layered treewidth $O(k+1)$, which are tight bounds that improve a previously known $O((k+1)^{3/4}n^{1/2})$ tr…
