scholarly journals On Bipartite Distinct Distances in the Plane

10.37236/9687 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Surya Mathialagan
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Crossing Number ◽  
Minimum Number

Given sets $\mathcal{P}, \mathcal{Q} \subseteq \mathbb{R}^2$ of sizes $m$ and $n$ respectively, we are interested in the number of distinct distances spanned by $\mathcal{P} \times \mathcal{Q}$. Let $D(m, n)$ denote the minimum number of distances determined by sets in $\mathbb{R}^2$ of sizes $m$ and $n$ respectively, where $m \leq n$. Elekes showed that $D(m, n) = O(\sqrt{mn})$ when $m \leqslant n^{1/3}$. For $m \geqslant n^{1/3}$, we have the upper bound $D(m, n) = O(n/\sqrt{\log n})$ as in the classical distinct distances problem.In this work, we show that Elekes' construction is tight by deriving the lower bound of $D(m, n) = \Omega(\sqrt{mn})$ when $m \leqslant n^{1/3}$. This is done by adapting Székely's crossing number argument. We also extend the Guth and Katz analysis for the classical distinct distances problem to show a lower bound of $D(m, n) = \Omega(\sqrt{mn}/\log n)$ when $m \geqslant n^{1/3}$.

scholarly journals NOTE ON THE NUMBER OF OBTUSE ANGLES IN POINT SETS

2014 ◽  
Vol 24 (03) ◽  
pp. 177-181 ◽  
Author(s):  
RUY FABILA-MONROY ◽  
CLEMENS HUEMER ◽  
EULÀLIA TRAMUNS
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
General Position ◽  
Crossing Number ◽  
The Other ◽  
Point Sets ◽  
Convex Position ◽  
Minimum Number

In 1979 Conway, Croft, Erdős and Guy proved that every set S of n points in general position in the plane determines at least [Formula: see text] obtuse angles and also presented a special set of n points to show the upper bound [Formula: see text] on the minimum number of obtuse angles among all sets S. We prove that every set S of n points in convex position determines at least [Formula: see text] obtuse angles, hence matching the upper bound (up to sub-cubic terms) in this case. Also on the other side, for point sets with low rectilinear crossing number, the lower bound on the minimum number of obtuse angles is improved.

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.

The multi-region index of a knot

2020 ◽  
Vol 29 (04) ◽  
pp. 2050022
Author(s):  
Sarah Goodhill ◽  
Adam M. Lowrance ◽  
Valeria Munoz Gonzales ◽  
Jessica Rattray ◽  
Amelia Zeh
Keyword(s):  
Lower Bound ◽  
Crossing Number ◽  
Number Of Generators ◽  
Branched Cover ◽  
Minimum Number

Using region crossing changes, we define a new invariant called the multi-region index of a knot. We prove that the multi-region index of a knot is bounded from above by twice the crossing number of the knot. In addition, we show that the minimum number of generators of the first homology of the double branched cover of [Formula: see text] over the knot is strictly less than the multi-region index. Our proof of this lower bound uses Goeritz matrices.

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.

ALTERNATING HAMILTON CYCLES WITH MINIMUM NUMBER OF CROSSINGS IN THE PLANE

2000 ◽  
Vol 10 (01) ◽  
pp. 73-78 ◽  
Author(s):  
ATSUSHI KANEKO ◽  
M. KANO ◽  
KIYOSHI YOSHIMOTO
Keyword(s):  
Upper Bound ◽  
Time Algorithm ◽  
Hamilton Cycle ◽  
Crossing Number ◽  
Hamilton Cycles ◽  
Line Segments ◽  
Straight Line ◽  
Minimum Number ◽  
Disjoint Sets

Let X and Y be two disjoint sets of points in the plane such that |X|=|Y| and no three points of X ∪ Y are on the same line. Then we can draw an alternating Hamilton cycle on X∪Y in the plane which passes through alternately points of X and those of Y, whose edges are straight-line segments, and which contains at most |X|-1 crossings. Our proof gives an O(n2 log n) time algorithm for finding such an alternating Hamilton cycle, where n =|X|. Moreover we show that the above upper bound |X|-1 on crossing number is best possible for some configurations.

scholarly journals On Compact Encoding of Pagenumber $k$ Graphs

2008 ◽  
Vol Vol. 10 no. 3 ◽  
Author(s):  
Cyril Gavoille ◽  
Nicolas Hanusse
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Constant Time ◽  
Worst Case ◽  
Encoding Scheme ◽  
Information Theoretic ◽  
Auxiliary Table ◽  
Minimum Number ◽  
International Audience

International audience In this paper we show an information-theoretic lower bound of kn - o(kn) on the minimum number of bits to represent an unlabeled simple connected n-node graph of pagenumber k. This has to be compared with the efficient encoding scheme of Munro and Raman of 2kn + 2m + o(kn+m) bits (m the number of edges), that is 4kn + 2n + o(kn) bits in the worst-case. For m-edge graphs of pagenumber k (with multi-edges and loops), we propose a 2mlog2k + O(m) bits encoding improving the best previous upper bound of Munro and Raman whenever m ≤ 1 / 2kn/log2 k. Actually our scheme applies to k-page embedding containing multi-edge and loops. Moreover, with an auxiliary table of o(m log k) bits, our coding supports (1) the computation of the degree of a node in constant time, (2) adjacency queries with O(logk) queries of type rank, select and match, that is in O(logk *minlogk / loglogm, loglogk) time and (3) the access to δ neighbors in O(δ) runs of select, rank or match;.

scholarly journals Crossings in Grid Drawings

10.37236/3025 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Vida Dujmović ◽  
Pat Morin ◽  
Adam Sheffer
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Geometric Graph ◽  
Upper Bounds ◽  
Crossing Number ◽  
Geometric Graphs ◽  
Vertex Sets

We prove tight crossing number inequalities for geometric graphs whose vertex sets are taken from a $d$-dimensional grid of volume $N$ and give applications of these inequalities to counting the number of crossing-free geometric graphs that can be drawn on such grids.In particular, we show that any geometric graph with $m\geq 8N$ edges and with vertices on a 3D integer grid of volume $N$, has $\Omega((m^2/N)\log(m/N))$ crossings. In $d$-dimensions, with $d\ge 4$, this bound becomes $\Omega(m^2/N)$. We provide matching upper bounds for all $d$. Finally, for $d\ge 4$ the upper bound implies that the maximum number of crossing-free geometric graphs with vertices on some $d$-dimensional grid of volume $N$ is $N^{\Theta(N)}$. In 3 dimensions it remains open to improve the trivial bounds, namely, the $2^{\Omega(N)}$ lower bound and the $N^{O(N)}$ upper bound.

A (5,5)-Colouring of Kn with Few Colours

2018 ◽  
Vol 27 (6) ◽  
pp. 892-912
Author(s):  
ALEX CAMERON ◽  
EMILY HEATH
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Minimum Number ◽  
Edge Colouring

For fixed integers p and q, let f(n,p,q) denote the minimum number of colours needed to colour all of the edges of the complete graph Kn such that no clique of p vertices spans fewer than q distinct colours. Any edge-colouring with this property is known as a (p,q)-colouring. We construct an explicit (5,5)-colouring that shows that f(n,5,5) ≤ n1/3 + o(1) as n → ∞. This improves upon the best known probabilistic upper bound of O(n1/2) given by Erdős and Gyárfás, and comes close to matching the best known lower bound Ω(n1/3).

scholarly journals An Explicit Edge-Coloring of $K_n$ with Six Colors on Every $K_5$

10.37236/7852 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Alex Cameron
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Edge Coloring ◽  
Minimum Number ◽  
Positive Integers

Let $p$ and $q$ be positive integers such that $1 \leq q \leq {p \choose 2}$. A $(p,q)$-coloring of the complete graph on $n$ vertices $K_n$ is an edge coloring for which every $p$-clique contains edges of at least $q$ distinct colors. We denote the minimum number of colors needed for such a $(p,q)$-coloring of $K_n$ by $f(n,p,q)$. This is known as the Erdös-Gyárfás function. In this paper we give an explicit $(5,6)$-coloring with $n^{1/2+o(1)}$ colors. This improves the best known upper bound of $f(n,5,6)=O\left(n^{3/5}\right)$ given by Erdös and Gyárfás, and comes close to matching the order of the best known lower bound, $f(n,5,6) = \Omega\left(n^{1/2}\right)$.

Some bounds on the minimum number of queries required for quantum channel perfect discrimination

2012 ◽  
Vol 12 (1&2) ◽  
pp. 138-148
Author(s):  
Cheng Lu ◽  
Jianxin Chen ◽  
Runyao Duan
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Quantum Channel ◽  
Necessary Condition ◽  
Quantum Channels ◽  
Sequential Scheme ◽  
Minimum Number ◽  
Perfect Discrimination ◽  
Sufficient And Necessary Condition

We prove a lower bound on the $q$-maximal fidelities between two quantum channels $\E_0$ and $\E_1$ and an upper bound on the $q$-maximal fidelities between a quantum channel $\E$ and an identity $\I$. Then we apply these two bounds to provide a simple sufficient and necessary condition for sequential perfect distinguishability between $\E$ and $\I$ and provide both a lower bound and an upper bound on the minimum number of queries required to sequentially perfectly discriminating $\E$ and $\I$. Interestingly, in the $2$-dimensional case, both bounds coincide. Based on the optimal perfect discrimination protocol presented in \cite{DFY09}, we can further generalize the lower bound and upper bound to the minimum number of queries to perfectly discriminating $\E$ and $I$ over all possible discrimination schemes. Finally the two lower bounds are shown remain working for perfectly discriminating general two quantum channels $\E_0$ and $\E_1$ in sequential scheme and over all possible discrimination schemes respectively.

Sign in / Sign up

Export Citation Format

Share Document