scholarly journals Number of connected spanning subgraphs on the Sierpinski gasket

2009 ◽  
Vol Vol. 11 no. 1 (Combinatorics) ◽  
Author(s):  
Shu-Chiuan Chang ◽  
Lung-Chi Chen
Keyword(s):  
Lower Bounds ◽  
Sierpinski Gasket ◽  
Upper And Lower Bounds ◽  
Sierpiński Gasket ◽  
Growth Constant ◽  
Asymptotic Growth ◽  
Spanning Subgraphs ◽  
International Audience

Combinatorics International audience We study the number of connected spanning subgraphs f(d,b) (n) on the generalized Sierpinski gasket SG(d,b) (n) at stage n with dimension d equal to two, three and four for b = 2, and layer b equal to three and four for d = 2. The upper and lower bounds for the asymptotic growth constant, defined as zSG(d,b) = lim(v ->infinity) ln f(d,b)(n)/v where v is the number of vertices, on SG(2,b) (n) with b = 2, 3, 4 are derived in terms of the results at a certain stage. The numerical values of zSG(d,b) are obtained.

scholarly journals Spanning forests on the Sierpinski gasket

2008 ◽  
Vol Vol. 10 no. 2 (Combinatorics) ◽  
Author(s):  
Shu-Chiuan Chang ◽  
Lung-Chi Chen
Keyword(s):  
Sierpinski Gasket ◽  
Upper Bounds ◽  
Sierpiński Gasket ◽  
Asymptotic Behaviors ◽  
Asymptotic Growth ◽  
Spanning Forests ◽  
International Audience ◽  
Growth Constants

Combinatorics International audience We study the number of spanning forests on the Sierpinski gasket SGd(n) at stage n with dimension d equal to two, three and four, and determine the asymptotic behaviors. The corresponding results on the generalized Sierpinski gasket SGd;b(n) with d = 2 and b = 3 ; 4 are obtained. We also derive upper bounds for the asymptotic growth constants for both SGd and SG2,b.

scholarly journals ACYCLIC ORIENTATIONS ON THE SIERPINSKI GASKET

2012 ◽  
Vol 26 (24) ◽  
pp. 1250128 ◽  
Author(s):  
SHU-CHIUAN CHANG
Keyword(s):  
Sierpinski Gasket ◽  
Upper Bounds ◽  
Two Dimensional ◽  
Sierpiński Gasket ◽  
Asymptotic Behaviors ◽  
Asymptotic Growth ◽  
Acyclic Orientations ◽  
Growth Constants

We study the number of acyclic orientations on the generalized two-dimensional Sierpinski gasket SG 2,b(n) at stage n with b equal to two and three, and determine the asymptotic behaviors. We also derive upper bounds for the asymptotic growth constants of SG 2,b and d-dimensional Sierpinski gasket SG d.

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 Structure of spanning trees on the two-dimensional Sierpinski gasket

2011 ◽  
Vol Vol. 12 no. 3 (Combinatorics) ◽  
Author(s):  
Shu-Chiuan Chang ◽  
Lung-Chi Chen
Keyword(s):  
Probability Distribution ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Sierpinski Gasket ◽  
Rational Numbers ◽  
Limiting Distribution ◽  
Two Dimensional ◽  
Sierpiński Gasket ◽  
Average Probability ◽  
International Audience

Combinatorics International audience Consider spanning trees on the two-dimensional Sierpinski gasket SG(n) where stage n is a non-negative integer. For any given vertex x of SG(n), we derive rigorously the probability distribution of the degree j ∈{1,2,3,4} at the vertex and its value in the infinite n limit. Adding up such probabilities of all the vertices divided by the number of vertices, we obtain the average probability distribution of the degree j. The corresponding limiting distribution φj gives the average probability that a vertex is connected by 1, 2, 3 or 4 bond(s) among all the spanning tree configurations. They are rational numbers given as φ1=10957/40464, φ2=6626035/13636368, φ3=2943139/13636368, φ4=124895/4545456.

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 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 Asymptotic Behavior of Boson Regge Trajectories

2021 ◽  
Vol 66 (2) ◽  
pp. 97
Author(s):  
A.A. Trushevsky
Keyword(s):  
Asymptotic Behavior ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Square Root ◽  
Asymptotic Growth ◽  
Regge Trajectories ◽  
Phase Representation

The asymptotic behavior of boson Regge trajectories is studied. Upper and lower bounds on the asymptotic growth of the trajectories are obtained using the phase representation for the trajectories and a number of physical requirements. It is shown that, within the assumptions made, the asymptotic behavior of the trajectories is a square root.

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.

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 $\textrm{disc}(\mathcal{H}^d,c) = O_{c,d}(\textrm{disc}(\mathcal{H},c)^d)$.

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