scholarly journals Discrepancy of Products of Hypergraphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Benjamin Doerr ◽  
Michael Gnewuch ◽  
Nils Hebbinghaus
Keyword(s):  
Direct Product ◽  
Lower Bounds ◽  
Research Paper ◽  
Arithmetic Progressions ◽  
Symmetric Product ◽  
Upper And Lower Bounds ◽  
Cartesian Products ◽  
International Audience ◽  
Better Than

International audience For a hypergraph $\mathcal{H} = (V,\mathcal{E})$, its $d$―fold symmetric product is $\Delta^d \mathcal{H} = (V^d,\{ E^d | E \in \mathcal{E} \})$. We give several upper and lower bounds for the $c$-color discrepancy of such products. In particular, we show that the bound $\textrm{disc}(\Delta^d \mathcal{H},2) \leq \textrm{disc}(\mathcal{H},2)$ proven for all $d$ in [B. Doerr, A. Srivastav, and P. Wehr, Discrepancy of Cartesian products of arithmetic progressions, Electron. J. Combin. 11(2004), Research Paper 5, 16 pp.] cannot be extended to more than $c = 2$ colors. In fact, for any $c$ and $d$ such that $c$ does not divide $d!$, there are hypergraphs having arbitrary large discrepancy and $\textrm{disc}(\Delta^d \mathcal{H},c) = \Omega_d(\textrm{disc}(\mathcal{H},c)^d)$. Apart from constant factors (depending on $c$ and $d$), in these cases the symmetric product behaves no better than the general direct product $\mathcal{H}^d$, which satisfies $\textrm{disc}(\mathcal{H}^d,c) = O_{c,d}(\textrm{disc}(\mathcal{H},c)^d)$.

scholarly journals Discrepancy of Symmetric Products of Hypergraphs

10.37236/1066 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Benjamin Doerr ◽  
Michael Gnewuch ◽  
Nils Hebbinghaus
Keyword(s):  
Direct Product ◽  
Lower Bounds ◽  
Research Paper ◽  
Arithmetic Progressions ◽  
Symmetric Product ◽  
Upper And Lower Bounds ◽  
Symmetric Products ◽  
Cartesian Products ◽  
Better Than

For a hypergraph ${\cal H} = (V,{\cal E})$, its $d$–fold symmetric product is defined to be $\Delta^d {\cal H} = (V^d,\{E^d |E \in {\cal E}\})$. We give several upper and lower bounds for the $c$-color discrepancy of such products. In particular, we show that the bound ${\rm disc}(\Delta^d {\cal H},2) \le {\rm disc}({\cal H},2)$ proven for all $d$ in [B. Doerr, A. Srivastav, and P. Wehr, Discrepancy of Cartesian products of arithmetic progressions, Electron. J. Combin. 11(2004), Research Paper 5, 16 pp.] cannot be extended to more than $c = 2$ colors. In fact, for any $c$ and $d$ such that $c$ does not divide $d!$, there are hypergraphs having arbitrary large discrepancy and ${\rm disc}(\Delta^d {\cal H},c) = \Omega_d({\rm disc}({\cal H},c)^d)$. Apart from constant factors (depending on $c$ and $d$), in these cases the symmetric product behaves no better than the general direct product ${\cal H}^d$, which satisfies ${\rm disc}({\cal H}^d,c) = O_{c,d}({\rm disc}({\cal H},c)^d)$.

scholarly journals On the history of van der Waerden's theorem on arithmetic progressions

2001 ◽  
Vol 32 (4) ◽  
pp. 335-341
Author(s):  
Tom C. Brown ◽  
Jau-Shyong Peter Shiue
Keyword(s):  
Lower Bounds ◽  
Arithmetic Progressions ◽  
Upper And Lower Bounds ◽  
Open Problems ◽  
History Of

In this expository note, we discuss the celebrated theorem known as ``van der Waerden's theorem on arithmetic progressions", the history of work on upper and lower bounds for the function associated with this theorem, a number of generalizations, and some open problems.

Tail of compound distributions and excess time

1996 ◽  
Vol 33 (01) ◽  
pp. 184-195 ◽  
Author(s):  
Xiaodong Lin
Keyword(s):  
Size Distribution ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Claim Size ◽  
Compound Distributions ◽  
Fundamental Identity ◽  
Claim Size Distribution ◽  
Simple Bounds ◽  
New Better Than Used ◽  
Better Than

Bounds on the tail of compound distributions are considered. Using a generalization of Wald's fundamental identity, we derive upper and lower bounds for various compound distributions in terms of new worse than used (NWU) and new better than used (NBU) distributions respectively. Simple bounds are obtained when the claim size distribution is NWUC, NBUC, NWU, NBU, IMRL, DMRL, DFR and IFR. Examples on how to use these bounds are given.

scholarly journals Monochromatic arithmetic progressions with large differences

1999 ◽  
Vol 60 (1) ◽  
pp. 21-35
Author(s):  
Tom C. Brown ◽  
Bruce M. Landman
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Arithmetic Progression ◽  
Constant Function ◽  
Arithmetic Progressions ◽  
Upper And Lower Bounds

A generalisation of the van der Waerden numbers w(k, r) is considered. For a function f: Z+ → R+ define w(f, k, r) to be the least positive integer (if it exists) such that for every r-coloring of [1, w(f, k, r)] there is a monochromatic arithmetic progression {a + id: 0 ≤ i ≤ k −1} such that d ≥ f(a). Upper and lower bounds are given for w(f, 3, 2). For k > 3 or r > 2, particular functions f are given such that w(f, k, r) does not exist. More results are obtained for the case in which f is a constant function.

scholarly journals On the connectedness and diameter of a geometric Johnson graph

2013 ◽  
Vol Vol. 15 no. 3 (Combinatorics) ◽  
Author(s):  
Crevel Bautista-Santiago ◽  
Javier Cano ◽  
Ruy Fabila-Monroy ◽  
David Flores-Peñaloza ◽  
Hernàn González-Aguilar ◽  
...  
Keyword(s):  
Lower Bounds ◽  
General Position ◽  
Convex Set ◽  
Upper And Lower Bounds ◽  
Johnson Graph ◽  
International Audience

Combinatorics International audience Let P be a set of n points in general position in the plane. A subset I of P is called an island if there exists a convex set C such that I = P \C. In this paper we define the generalized island Johnson graph of P as the graph whose vertex consists of all islands of P of cardinality k, two of which are adjacent if their intersection consists of exactly l elements. We show that for large enough values of n, this graph is connected, and give upper and lower bounds on its diameter.

scholarly journals Minimum survivable graphs with bounded distance increase

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Selma Djelloul ◽  
Mekkia Kouider
Keyword(s):  
Fault Tolerance ◽  
Lower Bounds ◽  
Interconnection Networks ◽  
Negative Integer ◽  
Upper And Lower Bounds ◽  
Minimum Size ◽  
Np Hard ◽  
Bounded Distance ◽  
Minimum Number ◽  
International Audience

International audience We study in graphs properties related to fault-tolerance in case a node fails. A graph G is k-self-repairing, where k is a non-negative integer, if after the removal of any vertex no distance in the surviving graph increases by more than k. In the design of interconnection networks such graphs guarantee good fault-tolerance properties. We give upper and lower bounds on the minimum number of edges of a k-self-repairing graph for prescribed k and n, where n is the order of the graph. We prove that the problem of finding, in a k-self-repairing graph, a spanning k-self-repairing subgraph of minimum size is NP-Hard.

scholarly journals On the enumeration of column-convex permutominoes

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Nicholas R. Beaton ◽  
Filippo Disanto ◽  
Anthony J. Guttmann ◽  
Simone Rinaldi
Keyword(s):  
Functional Equation ◽  
Asymptotic Behavior ◽  
Numerical Analysis ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Recursive Construction ◽  
Nous Obtenons ◽  
Comportement Asymptotique ◽  
Generating Trees ◽  
International Audience

International audience We study the enumeration of \emphcolumn-convex permutominoes, i.e. column-convex polyominoes defined by a pair of permutations. We provide a direct recursive construction for the column-convex permutominoes of a given size, based on the application of the ECO method and generating trees, which leads to a functional equation. Then we obtain some upper and lower bounds for the number of column-convex permutominoes, and conjecture its asymptotic behavior using numerical analysis. Nous étudions l'énumeration des \emphpermutominos verticalement convexes, c.à.d. les polyominos verticalement convexes définis par un couple de permutations. Nous donnons une construction recursive directe pour ces permutominos de taille fixée, basée sur une application de la méthode ECO et les arbres de génération, qui nous amène à une équat ion fonctionelle. Ensuite nous obtenons des bornes superieures et inférieures pour le nombre de ces permutominos convexes et nous conjecturons leur comportement asymptotique à l'aide d'analyses numériques.

scholarly journals Bounds on the sum of (log(p))2 Terms

2021 ◽  
Author(s):  
Jan Feliksiak
Keyword(s):  
Lower Bounds ◽  
Estimation Error ◽  
Research Paper ◽  
Point Of View ◽  
Upper And Lower Bounds ◽  
Log P ◽  
Fast Computation

In this research paper, we implement the theory of the primorial function, to develop the Supremum and Infimum bounds for the sum of (log(p))2. There are, however, considerable computational difficulties related to these bounds. Therefore, from a pragmatic point of view, a set of Upper and Lower bounds had been developed to bypass this issue. Despite the increased estimation error, the Upper and Lower bounds are still considered sufficiently accurate, while facilitating an easy and fast computation of the estimate of the sum.

scholarly journals Colouring the Square of the Cartesian Product of Trees

2011 ◽  
Vol Vol. 13 no. 2 (Graph and Algorithms) ◽  
Author(s):  
David R. Wood
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Cartesian Product ◽  
Upper And Lower Bounds ◽  
International Audience

Graphs and Algorithms International audience We prove upper and lower bounds on the chromatic number of the square of the cartesian product of trees. The bounds are equal if each tree has even maximum degree.

scholarly journals Optimal L(h,k)-Labeling of Regular Grids

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Tiziana Calamoneri
Keyword(s):  
Lower Bounds ◽  
Negative Integer ◽  
Upper And Lower Bounds ◽  
Vertex Coloring ◽  
Small Interval ◽  
Regular Grids ◽  
Special Cases ◽  
Hexagonal Grids ◽  
International Audience ◽  
The Difference

International audience The L(h, k)-labeling is an assignment of non negative integer labels to the nodes of a graph such that 'close' nodes have labels which differ by at least k, and 'very close' nodes have labels which differ by at least h. The span of an L(h,k)-labeling is the difference between the largest and the smallest assigned label. We study L(h, k)-labelings of cellular, squared and hexagonal grids, seeking those with minimum span for each value of k and h ≥ k. The L(h,k)-labeling problem has been intensively studied in some special cases, i.e. when k=0 (vertex coloring), h=k (vertex coloring the square of the graph) and h=2k (radio- or λ -coloring) but no results are known in the general case for regular grids. In this paper, we completely solve the L(h,k)-labeling problem on regular grids, finding exact values of the span for each value of h and k; only in a small interval we provide different upper and lower bounds.

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