On the Crossing Number of $K_{m,n}$

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) :Another long-standing conjecture states that the crossing number cr(K n ) of the complete graph K n equals Z(n) := 1 4. In this paper we show the following improved bounds on the asymptotic ratios of these crossing numbers and their conjectured values: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-ofthe-art quadratic optimization techniques combined with a bit of invariant theory of permutation groups to show that p ≥ 4.3593.

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“…A small improvement on the 0.8 factor (roughly to something around 0.8001) was recently reported by Nahas [15].…”

Section: Introductionmentioning

confidence: 51%

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

Klerk¹,

Maharry²,

Ṗasechnik³

et al. 2006

SIAM J. Discrete Math.

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“…The rectilinear drawing of a graph is defined as an embedding of it in R 2 such that its vertices are represented as points in general position (i.e., no three vertices are collinear) and edges are drawn as straight line segments connecting the corresponding vertices. The rectilinear crossing number of a graph G, denoted by cr(G), is defined as the minimum number of crossing pairs of edges among all rectilinear drawings of G. Determining the rectilinear crossing number of a graph is one of the most important problems in graph theory [1,8,9,12]. In particular, finding the rectilinear crossing numbers of complete bipartite graphs is an active area of research [9,13].…”

Section: Introductionmentioning

confidence: 99%

“…For any n ≥ 5, the bestknown lower and upper bounds on cr(K n,n ) are (n(n − 1)/5) n/2 (n − 1)/2 and n/2 2 (n − 1)/2 2 , respectively [9,16]. For sufficiently large n, the result of Nahas [12] improved the lower bound on cr(K n,n ) to (n(n − 1)/5) n/2 (n − 1)/2 + 9.9 × 10 −6 n 4 .…”

Section: Introductionmentioning

confidence: 99%

Rectilinear Crossings in Complete Balanced d-Partite d-Uniform Hypergraphs

Gangopadhyay

Shannigrahi

2020

Graphs and Combinatorics

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In this paper, we study the embedding of a complete balanced dpartite d-uniform hypergraph with its nd vertices represented as points in general position in R d and each hyperedge drawn as the convex hull of d corresponding vertices. We assume that the set of vertices is partitioned into d disjoint sets, each of size n, such that each vertex in a hyperedge is from a different set. Two hyperedges are said to be crossing if they are vertex disjoint and contain a common point in their relative interiors. Using Colored Tverberg theorem with restricted dimensions, we observe that such an embedding of a complete balanced d-partite d-uniform hypergraph with nd vertices contains Ω (8/3) d/2 (n/2) d ((n − 1)/2) d crossing pairs of hyperedges for n ≥ 3 and sufficiently large d. Using Gale transform and Ham-Sandwich theorem, we improve this lower bound to Ω 2 d (n/2) d ((n − 1)/2) d for n ≥ 3 and sufficiently large d.

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“…Computing the crossing number is N P-hard [40], and remains so for simple cubic graphs [46,65]. Moreover, the exact or even asymptotic crossing number is not known for specific graph families, such as complete graphs [72], complete bipartite graphs [55,70,72], and cartesian products [1,15,42,71].…”

Section: Introductionmentioning

confidence: 99%

“…Note that the assumption of bounded degree in Theorem 1.1 is unavoidable. For example, K 3,n has no K 5 -minor, yet has Ω(n 2 ) crossing number [70,55]. Conversely, bounded degree does not by itself guarantee linear crossing number.…”

Section: Introductionmentioning

confidence: 99%

Planar Decompositions and the Crossing Number of Graphs with an Excluded Minor

Wood

Telle

Graph Drawing

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Abstract. 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 the sense that a graph with bounded degree has linear crossing number if and only if it has a planar decomposition with bounded width and linear order. It follows from the above result about planar decompositions that every graph with bounded degree and an excluded minor has linear crossing number.Analogous results are proved for the convex and rectilinear crossing numbers. In particular, every graph with bounded degree and bounded tree-width has linear convex crossing number, and every K3,3-minor-free graph with bounded degree has linear rectilinear crossing number.

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On the Crossing Number of $K_{m,n}$

Cited by 12 publications

References 0 publications

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

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

Rectilinear Crossings in Complete Balanced d-Partite d-Uniform Hypergraphs

Planar Decompositions and the Crossing Number of Graphs with an Excluded Minor

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