Space Crossing Numbers

2012 ◽  
Vol 21 (3) ◽  
pp. 358-373 ◽  
Author(s):  
BORIS BUKH ◽  
ALFREDO HUBARD
Keyword(s):  
Lower Bound ◽  
Random Graphs ◽  
Crossing Number ◽  
Sharp Bounds ◽  
Crossing Numbers ◽  
Crossing Lemma ◽  
Rectilinear Space

We define a variant of the crossing number for an embedding of a graphGinto ℝ3, and prove a lower bound on it which almost implies the classical crossing lemma. We also give sharp bounds on the rectilinear space crossing numbers of pseudo-random graphs.

scholarly journals Braid index bounds ropelength from below

2020 ◽  
Vol 29 (04) ◽  
pp. 2050019
Author(s):  
Yuanan Diao
Keyword(s):  
Lower Bound ◽  
General Formula ◽  
Crossing Number ◽  
Crossing Numbers ◽  
Braid Index ◽  
Alternating Links

For an unoriented link [Formula: see text], let [Formula: see text] be the ropelength of [Formula: see text]. It is known that in general [Formula: see text] is at least of the order [Formula: see text], and at most of the order [Formula: see text] where [Formula: see text] is the minimum crossing number of [Formula: see text]. Furthermore, it is known that there exist families of (infinitely many) links with the property [Formula: see text]. A long standing open conjecture states that if [Formula: see text] is alternating, then [Formula: see text] is at least of the order [Formula: see text]. In this paper, we show that the braid index of a link also gives a lower bound of its ropelength. More specifically, we show that there exists a constant [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text] is the largest braid index among all braid indices corresponding to all possible orientation assignments of the components of [Formula: see text] (called the maximum braid index of [Formula: see text]). Consequently, [Formula: see text] for any link [Formula: see text] whose maximum braid index is proportional to its crossing number. In the case of alternating links, the maximum braid indices for many of them are proportional to their crossing numbers hence the above conjecture holds for these alternating links.

scholarly journals Crossing Numbers and Hard Erdős Problems in Discrete Geometry

1997 ◽  
Vol 6 (3) ◽  
pp. 353-358 ◽  
Author(s):  
LÁSZLÓ A. SZÉKELY
Keyword(s):  
Lower Bound ◽  
Discrete Geometry ◽  
Crossing Number ◽  
Plane Geometry ◽  
Crossing Numbers ◽  
Minimum Number

We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.

scholarly journals Multi-crossing number for knots and the Kauffman bracket polynomial

2016 ◽  
Vol 164 (1) ◽  
pp. 147-178 ◽  
Author(s):  
COLIN ADAMS ◽  
ORSOLA CAPOVILLA-SEARLE ◽  
JESSE FREEMAN ◽  
DANIEL IRVINE ◽  
SAMANTHA PETTI ◽  
...  
Keyword(s):  
Singular Point ◽  
Lower Bound ◽  
Crossing Number ◽  
Crossing Numbers ◽  
Kauffman Bracket ◽  
Classic Result ◽  
Formula Group ◽  
Image Position ◽  
Bracket Polynomial ◽  
Extensive List

AbstractA multi-crossing (or n-crossing) is a singular point in a projection of a knot or link at which n strands cross so that each strand bisects the crossing. We generalise the classic result of Kauffman, Murasugi and Thistlethwaite relating the span of the bracket polynomial to the double-crossing number of a link, span〈K〉 ⩽ 4c2, to the n-crossing number. We find the following lower bound on the n-crossing number in terms of the span of the bracket polynomial for any n ⩾ 3: $$\text{span} \langle K \rangle \leq \left(\left\lfloor\frac{n^2}{2}\right\rfloor + 4n - 8\right) c_n(K).$$ We also explore n-crossing additivity under composition, and find that for n ⩾ 4 there are examples of knots K1 and K2 such that cn(K1#K2) = cn(K1) + cn(K2) − 1. Further, we present the the first extensive list of calculations of n-crossing numbers of knots. Finally, we explore the monotonicity of the sequence of n-crossings of a knot, which we call the crossing spectrum.

scholarly journals Outerplanar Crossing Numbers, the Circular Arrangement Problem and Isoperimetric Functions

10.37236/1834 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Éva Czabarka ◽  
Ondrej Sýkora ◽  
László A. Székely ◽  
Imrich Vrťo
Keyword(s):  
Lower Bound ◽  
General Setting ◽  
Crossing Number ◽  
Linear Arrangement ◽  
Crossing Numbers ◽  
Arrangement Problem ◽  
Isoperimetric Functions ◽  
Circular Arrangement ◽  
Graph Families ◽  
Linear Arrangement Problem

We extend a lower bound due to Shahrokhi, Sýkora, Székely and Vrťo for the outerplanar crossing number (in other terminologies also called convex, circular and one-page book crossing number) to a more general setting. In this setting we can show a better lower bound for the outerplanar crossing number of hypercubes than the best lower bound for the planar crossing number. We exhibit further sequences of graphs, whose outerplanar crossing number exceeds by a factor of $\log n$ the planar crossing number of the graph. We study the circular arrangement problem, as a lower bound for the linear arrangement problem, in a general fashion. We obtain new lower bounds for the circular arrangement problem. All the results depend on establishing good isoperimetric functions for certain classes of graphs. For several graph families new near-tight isoperimetric functions are established.

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

10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
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 $n$.

scholarly journals Online Ramsey Games in Random Graphs

2009 ◽  
Vol 18 (1-2) ◽  
pp. 271-300 ◽  
Author(s):  
MARTIN MARCINISZYN ◽  
RETO SPÖHEL ◽  
ANGELIKA STEGER
Keyword(s):  
Lower Bound ◽  
Random Graphs ◽  
Empty Graph ◽  
Current Graph ◽  
Player Game ◽  
Monochromatic Copy

Consider the following one-player game. Starting with the empty graph onnvertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one ofravailable colours. The player's goal is to avoid creating a monochromatic copy of some fixed graphFfor as long as possible. We prove a lower bound ofnβ(F,r)on the typical duration of this game, where β(F,r) is a function that is strictly increasing inrand satisfies limr→∞β(F,r) = 2 − 1/m2(F), wheren2−1/m2(F)is the threshold of the corresponding offline colouring problem.

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

Mathematics ◽  
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.

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

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.

scholarly journals Minimal number of crossings in strong product of paths

2013 ◽  
Vol 29 (1) ◽  
pp. 27-32
Author(s):  
MARIAN KLESC ◽  
◽  
JANA PETRILLOVA ◽  
MATUS VALO ◽  
◽  
...  
Keyword(s):  
Crossing Number ◽  
Cartesian Products ◽  
Crossing Numbers ◽  
Strong Product ◽  
Product Of Graphs

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.

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