Cyclic‐order graphs and Zarankiewicz's crossing‐number conjecture

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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“…This inequality is stated in [11] and proved in [24]. This observation alone yields a lower bound for cr(K m,n ), as follows.…”

Section: Introductionmentioning

confidence: 67%

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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“…This graph is represented by a square symmetrical adjacency matrix. The distance between each pair of vertices are calculated using the properties of the cyclicorder graph CO 5 defined in [5].…”

Section: Software Descriptionmentioning

confidence: 99%

ON THE CROSSING NUMBER OF THE JOIN OF FIVE VERTEX GRAPH WITH THE DISCRETE GRAPH Dn

Berežný¹,

Staš²

2017

AEI

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In this paper, we show the values of crossing numbers for join products of graph G on five vertices with the discrete graph D n and the path P n on n vertices. The proof is done with the help of software. The software generates all cyclic permutations for a given number n. For cyclic permutations, P 1 -P m will create a graph in which to calculate the distances between all vertices of the graph. These distances are used in proof of crossing numbers of presented graphs.

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“…In [13], Kleitman established the crossing number of the latter when (i) or (ii) apply, and, in [29], Woodall established it under conditions (iii) or (iv). In each of the cases, we can add the edges of the cycle into an optimal drawing of K n,d from [28] without introducing new crossings, cf.…”

Section: Moreover Under the Above Conditions There Exist Pairwisementioning

confidence: 97%

On the crossing numbers of Cartesian products with trees

Bokal¹

2007

Journal of Graph Theory

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Abstract:Zip product was recently used in a note establishing the crossing number of the Cartesian product K 1,n P m . In this article, we further investigate the relations of this graph operation with the crossing numbers of graphs. First, we use a refining of the embedding method bound for crossing numbers to weaken the connectivity condition under which the crossing number is additive for the zip product. Next, we deduce a general theorem for bounding the crossing numbers of (capped) Cartesian product of graphs with trees, which yields exact results under certain symmetry conditions. We apply this theorem to obtain exact and approximate results on crossing numbers of Cartesian product of various graphs with trees.

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Cyclic‐order graphs and Zarankiewicz's crossing‐number conjecture

Cited by 87 publications

References 7 publications

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

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

ON THE CROSSING NUMBER OF THE JOIN OF FIVE VERTEX GRAPH WITH THE DISCRETE GRAPH Dn

On the crossing numbers of Cartesian products with trees

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