Crossing Number Cr Research Articles

The influence of separating cycles in drawings of K 5 ∖ e in the join product with paths and cycles

Abstract The crossing number cr(H) of a graph H is the minimum number of edge crossings over all drawings of H in the plane. Let H ∗ be the connected graph of order five isomorphic to K 5 ∖ e obtained by removing one edge from the complete graph K 5. The main aim of the paper is to give the crossing numbers of the join products H ∗ + Pn and H ∗ + Cn , where Pn and Cn are the path and the cycle on n vertices, respectively. The proofs are done with the help of a suitable classification of a large number of drawings of the graph H ∗ in view of the existence of a separating cycle of two possible types.

Crossing Numbers of the Cartesian Product of the Double Triangular Snake Graphs With Path Pm.

The crossing number Cr(G) of a graph G is the least number of edge crossings in all possible good drawings of G in the plane. Join and Cartesian products of graphs have many interesting graph-theoretical properties. In this paper, we evaluate the crossing number of the Cartesian product of double triangular snake graph DT2 with the path Pm. In this paper, we proved Cr(DT2 × Pm) = 6(m − 2), form ≥ 2 where Cr denotes the crossing number

Disconnected spanning subgraphs of paths in the join products with cycles

The crossing number cr(G) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main aim of this paper is to give the crossing numbers of the join products of the cycles Cn on n vertices with all nonempty disconnected spanning subgraphs of the paths P5 and P6 on five and six vertices, respectively. Recently, this problem was solved for the nonempty disconnected spanning subgraphs of the paths on three and four vertices by Klešč.

Crossing numbers of complete bipartite graphs

The long standing Zarankiewicz's conjecture states that the crossing number cr(Km,n) of the complete bipartite graph is Z(m,n):= [m/2][m-1/2][n/2][n-1/2]. Using flag algebras we show that cr(Kn,n) ≥ 0.9118 • Z(n, n) + o(n4). We also show that the rectilinear crossing number cr-(Kn,n) of Kn,n is at least 0.987 • Z(n,n) + o(n4). Finally, we show that if a drawing of Kn,n has no K3,4 that has exactly two crossings, and these crossings share exactly one vertex, then it has at least Z(n,n) + o(n4) crossings. This is a local restriction inspired by Turán type problems that gives an asymptotically tight result.

Semimeander Crossing Number of Knots and Related Invariants

The minimum number of crossings among all of the diagrams of a knot K composed of at most k smooth simple arcs is called the k-arc crossing number of K. This number is denoted by crk(K). The 2-arc crossing number is also called the semimeander crossing number. The article studies connections of the k-arc crossing numbers with the classical crossing number cr(K) of K. It is proved that for each knot K, the following inequalities are fulfilled: $$ {\mathrm{cr}}_2(K)\le {\sqrt[4]{6}}^{\mathrm{cr}(K)} $$ and $$ {\mathrm{cr}}_k(K)\le {\mathrm{cr}}_{k+1}(K)+2\frac{{\left({\mathrm{cr}}_{k+1}(K)\right)}^2}{{\left(k+1\right)}^2}. $$

Join Products K2,3 + Cn

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

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 $0.996\, H(n)$.

On Crossing Numbers of Complete Tripartite and Balanced Complete Multipartite Graphs

AbstractThe crossing number cr(G) of a graph G is the minimum number of crossings in a drawing of G in the plane with no more than two edges intersecting at any point that is not a vertex. The rectilinear crossing number of G is the minimum number of crossings in a such drawing of G with edges as straight line segments. Zarankiewicz proved in 1952 that . We generalize the upper bound to urn:x-wiley:03649024:media:jgt22041:jgt22041-math-0004and prove . We also show that for n large enough, and , with the tighter rectilinear lower bound established through the use of flag algebras. A complete multipartite graph is balanced if the partite sets all have the same cardinality. We study asymptotic behavior of the crossing number of the balanced complete r‐partite graph. Richter and Thomassen proved in 1997 that the limit as of over the maximum number of crossings in a drawing of exists and is at most . We define and show that for a fixed r and the balanced complete r‐partite graph, is an upper bound to the limit superior of the crossing number divided by the maximum number of crossings in a drawing.

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

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 $r,s$.

Minimal number of crossings in strong product of paths

The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. The exact crossing number is known only for few specific families of graphs. Cartesian products of two graphs belong to the first families of graphs for which the crossing number has been studied. Some results concerning crossing numbers are also known for join products of two graphs. In the paper, we start to collect the crossing numbers for the strong product of graphs, namely for the strong product of two paths.

On the crossing numbers of Cartesian products of stars and graphs of order six

The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. According to their special structure, the class of Cartesian products of two graphs is one of few graph classes for which some exact values of crossing numbers were obtained. The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. Moreover, except of six graphs, the crossing numbers of Cartesian products G K1,n for all other connected graphs G on five vertices are known. In this paper we are dealing with the Cartesian products of stars with graphs on six vertices. We give the exact values of crossing numbers for some of these graphs and we summarise all known results concerning crossing numbers of these graphs. Moreover, we give the crossing number of G1 T for the special graph G1 on six vertices and for any tree T with no vertex of degree two as well as the crossing number of K1,n T for any tree T with maximum degree five.

Zarankiewiczʼs Conjecture is finite for each fixed m

Zarankiewiczʼs Crossing Number Conjecture states that the crossing number cr(Km,n) of the complete bipartite graph Km,n equals Z(m,n):=⌊m/2⌋⌊(m−1)/2⌋⌊n/2⌋⌊(n−1)/2⌋, for all positive integers m, n. This conjecture has only been verified for min{m,n}⩽6, for K7,7, K7,8, K7,9, and K7,10, and for K8,8, K8,9, and K8,10. We determine, for each positive integer m, an integer N0=N0(m) with the following property: if cr(Km,n)=Z(m,n) for all n⩽N0, then cr(Km,n)=Z(m,n) for every n. This yields, for each fixed integer m, a finite algorithm that either shows that the assertion: “for all n, cr(Km,n)=Z(m,n)” is true, or else finds a counterexample.

Asymptotically settling Zarankiewiczʼs Conjecture in finite time, for each m

Abstract Perhaps the most notorious open problem in crossing numbers is Zarankiewiczʼs Conjecture, which states that the crossing number cr ( K m , n ) of the complete bipartite graph cr ( K m , n ) is Z ( m , n ) : = ⌊ m − 1 2 ⌋ ⌊ m 2 ⌋ ⌊ n − 1 2 ⌋ ⌊ n 2 ⌋ . This has been verified only for min { m , n } ⩽ 6 and for a few special cases. We have proved that, for each m, there is an integer N ( m ) with the following property: if cr ( K m , n ) = Z ( m , n ) for all n ⩽ N ( m ) , then cr ( K ) = Z ( m , n ) for all n. This yields, for each fixed m, an algorithm that either decides that Zarankiewiczʼs Conjeture cr ( K m , n ) = Z ( m , n ) is true for all n, or else finds a counterexample. To illustrate our techniques, we consider the Asymptotic Zarankiewiczʼs Conjecture (let m be a fixed positive integer; then lim n → ∞ cr ( K m , n ) / Z ( m , n ) = 1 ). We give a detailed sketch of the proof that the Asymptotic Zarankiewiczʼs Conjecture can be settled in finite time.

The crossing numbers of join products of paths with graphs of order four

Kulli and Muddebihal [V.R. Kulli, M.H. Muddebihal, Characterization of join graphs with crossing number zero, Far East J. Appl. Math. 5 (2001) 87–97] gave the characterization of all pairs of graphs which join product is planar graph. The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. There are only few results concerning crossing numbers of graphs obtained as join product of two graphs. In the paper, the exact values of crossing numbers for join of paths with all graphs of order four, as well as for join of all graphs of order four with n isolated vertices are given.

On the decay of crossing numbers

The crossing number cr ( G ) of a graph G is the minimum number of crossings over all drawings of G in the plane. In 1993, Richter and Thomassen [B. Richter, C. Thomassen, Minimal graphs with crossing number at least k, J. Combin. Theory Ser. B 58 (1993) 217–224] conjectured that there is a constant c such that every graph G with crossing number k has an edge e such that cr ( G − e ) ⩾ k − c k . They showed only that G always has an edge e with cr ( G − e ) ⩾ 2 5 cr ( G ) − O ( 1 ) . We prove that for every fixed ϵ > 0 , there is a constant n 0 depending on ϵ such that if G is a graph with n > n 0 vertices and m > n 1 + ϵ edges, then G has a subgraph G ′ with at most ( 1 − ϵ 24 ) m edges such that cr ( G ′ ) ⩾ ( 1 28 − o ( 1 ) ) cr ( G ) .

Geometric drawings of [formula omitted] with few crossings

We give a new upper bound for the rectilinear crossing number cr ¯ ( n ) of the complete geometric graph K n . We prove that cr ¯ ( n ) ⩽ 0.380559 ( n 4 ) + Θ ( n 3 ) by means of a new construction based on an iterative duplication strategy starting with a set having a certain structure of halving lines.

K-Sets, Convex Quadrilaterals, and the Rectilinear Crossing Number of Kn

We use circular sequences to give an improved lower bound on the minimum number of (≤ k)-sets in a set of points in general position. We then use this to show that if S is a set of n points in general position, then the number □(S) of convex quadrilaterals determined by the points in S is at least 0.37553(n4) + O(n3). This in turn implies that the rectilinear crossing number cr(Kn) of the complete graph Knis at least 0.37553(n4) + O(n3), and that Sylvester's Four Point Problem Constant is at least 0.37553. These improved bounds refine results recently obtained by Abrego and Fernandez-Merchant and by Lovasz, Vesztergombi, Wagner, and Welzl.

Improved Bounds for the Crossing Numbers of Km,n and Kn

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.

An improvement of the crossing number bound

AbstractThe crossing number cr(G) of a simple graph G with n vertices and m edges is the minimum number of edge crossings over all drawings of G on the ℝ2 plane. The conjecture made by Erdős in 1973 that cr(G) ≥ Cm3/n2 was proved in 1982 by Leighton with C = 1/100 and this constant was gradually improved to reach the best known value C = 1/31.08 obtained recently by Pach, Radoicčić, Tardos, and Tóth [4] for graphs such that m ≥ 103n/16. We improve this result with values for the constant in the range 1/31.08 ≤ C &< 1/15 where C depends on m/n2. For example, C > 1/25 for graphs with m/n2 > 0.291 and n > 22, and C > 1/20 for dense graphs with m/n2 ≥ 0.485. © 2005 Wiley Periodicals, Inc. J Graph Theory

Crossing number, pair-crossing number, and expansion

The crossing number cr(G) of a graph G is the minimum possible number of edge crossings in a drawing of G in the plane, while the pair-crossing number pcr(G) is the smallest number of pairs of edges that cross in a drawing of G in the plane. While cr(G)⩾pcr(G) holds trivially, it is not known whether a strict inequality can ever occur (this question was raised by Mohar and Pach and Tóth). We aim at bounding cr(G) in terms of pcr(G). Using the methods of Leighton and Rao, Bhatt and Leighton, and Even, Guha and Schieber, we prove that cr(G)=O(log3n(pcr(G)+ssqd(G))), where n=|V(G)| and ssqd(G)=∑v∈V(G)degG(v)2. One of the main steps is an analogy of the well-known lower bound cr(G)=Ω(b(G)2)−O(ssqd(G)), where b(G) is the bisection width of G, that is, the smallest number of edges that have to be removed so that no component of the resulting graph has more than 23n vertices. We show that pcr(G)=Ω(b(G)2/log2n)−O(ssqd(G)).We also prove by similar methods that a graph G with crossing number k=cr(G)>Cssqd(G)mlog2n has a nonplanar subgraph on at most OΔnmlog2nk vertices, where m is the number of edges, Δ is the maximum degree in G, and C is a suitable sufficiently large constant.