Average crossing number and writhe of knotted random polygons in confinement

The average crossing number of equilateral random polygons

Journal of Physics A General Physics ◽

10.1088/0305-4470/36/46/002 ◽

2003 ◽

Vol 36 (46) ◽

pp. 11561-11574 ◽

Cited By ~ 31

Author(s):

Y Diao ◽

A Dobay ◽

R B Kusner ◽

K Millett ◽

A Stasiak

Keyword(s):

Crossing Number ◽

Random Polygons

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

The Electronic Journal of Combinatorics ◽

10.37236/1748 ◽

2003 ◽

Vol 10 (1) ◽

Cited By ~ 4

Author(s):

Nagi H. Nahas

Keyword(s):

Lower Bound ◽

Bipartite Graph ◽

Complete Bipartite Graph ◽

Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

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THE CYLINDRICAL CROSSING NUMBER OF ZERO DIVISOR GRAPHS

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.8.57 ◽

2020 ◽

Vol 9 (8) ◽

pp. 5901-5908

Author(s):

M. Sagaya Nathan ◽

J. Ravi Sankar

Keyword(s):

Zero Divisor ◽

Crossing Number

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Join Products K2,3 + Cn

Mathematics ◽

10.3390/math8060925 ◽

2020 ◽

Vol 8 (6) ◽

pp. 925

Author(s):

Michal Staš

Keyword(s):

Bipartite Graph ◽

Complete Bipartite Graph ◽

Crossing Number ◽

Arithmetic Means ◽

Minimum Number ◽

Edge Crossings

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 .

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Improvement on the Crossing Number of Crossing-Critical Graphs

Discrete & Computational Geometry ◽

10.1007/s00454-020-00264-2 ◽

2020 ◽

Author(s):

János Barát ◽

Géza Tóth

Keyword(s):

Crossing Number ◽

Critical Graphs ◽

Minimum Number ◽

Edge Crossings

AbstractThe 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 $$2.5\, k+16$$ 2.5 k + 16 . We improve this bound to $$2k+8\sqrt{k}+47$$ 2 k + 8 k + 47 .

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An Improvement of the Lower Bound on the Minimum Number of ≤k-Edges

Mathematics ◽

10.3390/math9050525 ◽

2021 ◽

Vol 9 (5) ◽

pp. 525

Author(s):

Javier Rodrigo ◽

Susana Merchán ◽

Danilo Magistrali ◽

Mariló López

Keyword(s):

Lower Bound ◽

Complete Graph ◽

General Position ◽

Crossing Number ◽

Minimum Number

In this paper, we improve the lower bound on the minimum number of ≤k-edges in sets of n points in general position in the plane when k is close to n2. As a consequence, we improve the current best lower bound of the rectilinear crossing number of the complete graph Kn for some values of n.

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Stick numbers of Montesinos knots

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216521500139 ◽

2021 ◽

pp. 2150013

Author(s):

Hwa Jeong Lee ◽

Sungjong No ◽

Seungsang Oh

Keyword(s):

Upper Bound ◽

Crossing Number ◽

Number Formula ◽

Knots And Links

Negami found an upper bound on the stick number [Formula: see text] of a nontrivial knot [Formula: see text] in terms of the minimal crossing number [Formula: see text]: [Formula: see text]. Huh and Oh found an improved upper bound: [Formula: see text]. Huh, No and Oh proved that [Formula: see text] for a [Formula: see text]-bridge knot or link [Formula: see text] with at least six crossings. As a sequel to this study, we present an upper bound on the stick number of Montesinos knots and links. Let [Formula: see text] be a knot or link which admits a reduced Montesinos diagram with [Formula: see text] crossings. If each rational tangle in the diagram has five or more index of the related Conway notation, then [Formula: see text]. Furthermore, if [Formula: see text] is alternating, then we can additionally reduce the upper bound by [Formula: see text].

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Cyclic permutations and crossing numbers of join products of two symmetric graphs of order six

Carpathian Journal of Mathematics ◽

10.37193/cjm.2019.02.02 ◽

2019 ◽

Vol 35 (2) ◽

pp. 137-146

Author(s):

STEFAN BEREZNY ◽

MICHAL STAS ◽

◽

Keyword(s):

Crossing Number ◽

Crossing Numbers ◽

Symmetric Graphs ◽

Isolated Vertex ◽

Join Of Graphs ◽

Discrete Graph

The main purpose of this article is broaden known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product G + Dn, where the graph G consists of one 5-cycle and of one isolated vertex, and Dn consists on n isolated vertices. The proof is done with the help of software that generates all cyclic permutations for a given number k, and creates a new graph COG for calculating the distances between all vertices of the graph. Finally, by adding some edges to the graph G, we are able to obtain the crossing numbers of the join product with the discrete graph Dn and with the path Pn on n vertices for other two graphs.

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