The crossing number of the Cartesian product of paths with complete graphs

Ouyang, Zhangdong; Wang, Jing; Huang, Yuanqiu

doi:10.1016/j.disc.2014.03.011

Cited by 10 publications

(6 citation statements)

References 22 publications

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“…The following result indicates that the crossing number is also additive for zip product under the condition of being 4-strict-star-connected. Lemma 9 (Ouyang et al [22]). Let G 1 and G 2 be disjoint graphs,…”

Section: Corollarymentioning

confidence: 99%

“…In this section, we establish some general theorems for bounding the crossing number of (capped) Cartesian product of graphs with trees, which may yield exact results under certain conditions. These results are obtained by applying Theorem 2 in the previous section and Lemma 4.1 in [22]. The technique is similar to that of [5].…”

Section: Bounds For the Crossing Number Of Cartesian Product With Treesmentioning

confidence: 99%

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Zip product of graphs and crossing numbers

Ouyang

Huang

Dong

et al. 2020

Journal of Graph Theory

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D. Bokal proved that the crossing number is additive for the zip product under the condition of having two coherent bundles in the zipped graphs. This property is very effective when dealing with the crossing numbers of (capped) Cartesian product of trees with graphs containing a dominating vertex. In this paper, we first prove that the crossing number is still additive for the zip product under a weaker condition. Based on the new condition, we then establish some general expressions for bounding the crossing numbers of (capped) Cartesian product of trees with graphs (possibly without dominating vertex). Exact values of the crossing numbers of Cartesian product of trees with most graphs of order at most five are obtained by applying these expressions, which extend some previous results due to M. Klešč. In fact, our results can also be applied to deal with Cartesian product of trees with graphs of order more than five.

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Section: Corollarymentioning

confidence: 99%

Section: Bounds For the Crossing Number Of Cartesian Product With Treesmentioning

confidence: 99%

Zip product of graphs and crossing numbers

Ouyang

Huang

Dong

et al. 2020

Journal of Graph Theory

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“…In It should be noted that Chia and Lee [19] also discovered a result equivalent to [146], however it did not appear until 2015 due to being published in Ars Combinatoria, which has a notoriously large backlog of publications.…”

Section: Complete Graphs Minus An Edgementioning

confidence: 99%

“…2014, [146] proved that Conjecture 2.26 holds for even n whenever Conjecture 2.2 holds for n − 1. Hence, Conjecture 2.26 is currently known to hold for n ≤ 12.…”

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confidence: 99%

A survey of graphs with known or bounded crossing numbers

Clancy¹,

Haythorpe²,

Newcombe³

2019

Preprint

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“…Let P n and C n denote the path and cycle on n vertices respectively. By an m-prism, denoted PðmÞ, we mean the Cartesian product C m w P 2 : The crossing numbers of the Cartesian product of some special graphs with the path (or other graph) have been the subject of investigation (see [2][3][4][5][6][7][8][9][10][11][12][13]). In particular, in [7], it was proved that crðPð3Þ w P n Þ ¼ 4ðn À 1Þ for all natural numbers n !…”

Section: Introductionmentioning

confidence: 99%

Crossing number of Cartesian product of prism and path

Yiew

Chia

Ong

2020

AKCE International Journal of Graphs and Combinatorics

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An m-prism is the Cartesian product of an m-cycle and a path with 2 vertices. We prove that the crossing number of the join of an m-prism (m ! 4) and a graph with k isolated vertices is km for each k 2 f1, 2g: We then use this result to prove that the crossing number of the Cartesian product of a 5-prism and a path with n vertices is 10ðn À 1Þ: This answers partially the conjecture raised by Peng and Yiew (in 2006) in the affirmative.

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The crossing number of the Cartesian product of paths with complete graphs

Cited by 10 publications

References 22 publications

Zip product of graphs and crossing numbers

Zip product of graphs and crossing numbers

A survey of graphs with known or bounded crossing numbers

Crossing number of Cartesian product of prism and path

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