scholarly journals Optimal Lower Bound for 2-Identifying Codes in the Hexagonal Grid

10.37236/2414 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Ville Junnila ◽  
Tero Laihonen
Keyword(s):  
Lower Bound ◽  
Hexagonal Grid ◽  
Identifying Codes ◽  
Identifying Code ◽  
Optimal Lower Bound

An $r$-identifying code in a graph $G = (V,E)$ is a subset $C \subseteq V$ such that for each $u \in V$ the intersection of $C$ and the ball of radius $r$ centered at $u$ is non-empty and unique. Previously, $r$-identifying codes have been studied in various grids. In particular, it has been shown that there exists a $2$-identifying code in the hexagonal grid with density $4/19$ and that there are no $2$-identifying codes with density smaller than $2/11$. Recently, the lower bound has been improved to $1/5$ by Martin and Stanton (2010). In this paper, we prove that the $2$-identifying code with density $4/19$ is optimal, i.e. that there does not exist a $2$-identifying code in the hexagonal grid with smaller density.

scholarly journals A New Lower Bound on the Density of Vertex Identifying Codes for the Infinite Hexagonal Grid

10.37236/202 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Daniel W. Cranston ◽  
Gexin Yu
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Hexagonal Grid ◽  
Identifying Codes ◽  
Identifying Code ◽  
Vertex Set

Given a graph $G$, an identifying code ${\cal D}\subseteq V(G)$ is a vertex set such that for any two distinct vertices $v_1,v_2\in V(G)$, the sets $N[v_1]\cap{\cal D}$ and $N[v_2]\cap{\cal D}$ are distinct and nonempty (here $N[v]$ denotes a vertex $v$ and its neighbors). We study the case when $G$ is the infinite hexagonal grid $H$. Cohen et.al. constructed two identifying codes for $H$ with density $3/7$ and proved that any identifying code for $H$ must have density at least $16/39\approx0.410256$. Both their upper and lower bounds were best known until now. Here we prove a lower bound of $12/29\approx0.413793$.

scholarly journals Lower Bounds for Identifying Codes in some Infinite Grids

10.37236/394 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Ryan Martin ◽  
Brendon Stanton
Keyword(s):  
Lower Bounds ◽  
Finite Graph ◽  
Infinite Graphs ◽  
Locally Finite ◽  
Identifying Codes ◽  
Identifying Code ◽  
Hexagonal Grids ◽  
Closed Neighborhood ◽  
Definition Of ◽  
Infinite Grids

An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and unique. On a finite graph, the density of a code is $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We present new lower bounds for densities of codes for some small values of $r$ in both the square and hexagonal grids.

scholarly journals General Bounds for Identifying Codes in Some Infinite Regular Graphs

10.37236/1583 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Irène Charon ◽  
Iiro Honkala ◽  
Olivier Hudry ◽  
Antoine Lobstein
Keyword(s):  
Undirected Graph ◽  
Upper Bounds ◽  
Regular Graphs ◽  
Lower And Upper Bounds ◽  
Identifying Codes ◽  
Identifying Code

Consider a connected undirected graph $G=(V,E)$ and a subset of vertices $C$. If for all vertices $v \in V$, the sets $B_r(v) \cap C$ are all nonempty and pairwise distinct, where $B_r(v)$ denotes the set of all points within distance $r$ from $v$, then we call $C$ an $r$-identifying code. We give general lower and upper bounds on the best possible density of $r$-identifying codes in three infinite regular graphs.

On the Inf-Sup Constant of a Triangular Spectral Method for the Stokes Equations

2016 ◽  
Vol 16 (3) ◽  
pp. 507-522 ◽  
Author(s):  
Yanhui Su ◽  
Lizhen Chen ◽  
Xianjuan Li ◽  
Chuanju Xu
Keyword(s):  
Lower Bound ◽  
Error Estimate ◽  
Spectral Method ◽  
Stokes Equations ◽  
Necessary Condition ◽  
Well Posedness ◽  
Saddle Point Problems ◽  
Numerical Examples ◽  
Lbb Condition ◽  
Optimal Lower Bound

AbstractThe Ladyženskaja–Babuška–Brezzi (LBB) condition is a necessary condition for the well-posedness of discrete saddle point problems stemming from discretizing the Stokes equations. In this paper, we prove the LBB condition and provide the (optimal) lower bound for this condition for the triangular spectral method proposed by L. Chen, J. Shen, and C. Xu in [3]. Then this lower bound is used to derive an error estimate for the pressure. Some numerical examples are provided to confirm the theoretical estimates.

An optimal lower bound for nonregular languages

1994 ◽  
Vol 52 (6) ◽  
pp. 339 ◽  
Author(s):  
A. Bertoni ◽  
Carlo Mereghetti ◽  
Giovanni Pighizzini
Keyword(s):  
Lower Bound ◽  
Optimal Lower Bound

scholarly journals Extreme Values of the Riemann Zeta Function on the 1-Line

2017 ◽  
Vol 2019 (22) ◽  
pp. 6924-6932 ◽  
Author(s):  
Christoph Aistleitner ◽  
Kamalakshya Mahatab ◽  
Marc Munsch
Keyword(s):  
Lower Bound ◽  
Extreme Values ◽  
Mean Square ◽  
Self Similarity ◽  
New Variant ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
The Mean ◽  
Resonator Method ◽  
Optimal Lower Bound

Abstract We prove that there are arbitrarily large values of t such that $|\zeta (1+it)| \geq e^{\gamma } (\log _{2} t +\log _{3} t) + \mathcal{O}(1)$. This essentially matches the prediction for the optimal lower bound in a conjecture of Granville and Soundararajan. Our proof uses a new variant of the “long resonator” method. While earlier implementations of this method crucially relied on a “sparsification” technique to control the mean-square of the resonator function, in the present paper we exploit certain self-similarity properties of a specially designed resonator function.

An optimal lower bound on the number of variables for graph identification

COMBINATORICA ◽  
1992 ◽  
Vol 12 (4) ◽  
pp. 389-410 ◽  
Author(s):  
Jin-Yi Cai ◽  
Martin F�rer ◽  
Neil Immerman
Keyword(s):  
Lower Bound ◽  
Optimal Lower Bound
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