scholarly journals Cycle-Saturated Graphs with Minimum Number of Edges

2012 ◽  
Vol 73 (2) ◽  
pp. 203-215 ◽  
Author(s):  
Zoltán Füredi ◽  
Younjin Kim
Keyword(s):  
Saturated Graphs ◽  
Minimum Number

scholarly journals Saturation Numbers for Trees

10.37236/180 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Jill Faudree ◽  
Ralph J. Faudree ◽  
Ronald J. Gould ◽  
Michael S. Jacobson
Keyword(s):  
Saturated Graphs ◽  
Minimum Number ◽  
Saturation Number

For a fixed graph $H$, a graph $G$ is $H$-saturated if there is no copy of $H$ in $G$, but for any edge $e \notin G$, there is a copy of $H$ in $G + e$. The collection of $H$-saturated graphs of order $n$ is denoted by ${\bf SAT}(n,H)$, and the saturation number, ${\bf sat}(n, H),$ is the minimum number of edges in a graph in ${\bf SAT}(n,H)$. Let $T_k$ be a tree on $k$ vertices. The saturation numbers ${\bf sat}(n,T_k)$ for some families of trees will be determined precisely. Some classes of trees for which ${\bf sat}(n, T_k) < n$ will be identified, and trees $T_k$ in which graphs in ${\bf SAT}(n,T_k)$ are forests will be presented. Also, families of trees for which ${\bf sat}(n,T_k) \geq n$ will be presented. The maximum and minimum values of ${\bf sat}(n,T_k)$ for the class of all trees will be given. Some properties of ${\bf sat}(n,T_k)$ and ${\bf SAT} (n,T_k)$ for trees will be discussed.

scholarly journals Constructive Upper Bounds for Cycle-Saturated Graphs of Minimum Size

10.37236/1055 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Ronald Gould ◽  
Tomasz Łuczak ◽  
John Schmitt
Keyword(s):  
Upper Bounds ◽  
Minimum Size ◽  
Saturated Graphs ◽  
Minimum Number

A graph $G$ is said to be $C_l$-saturated if $G$ contains no cycle of length $l$, but for any edge in the complement of $G$ the graph $G+e$ does contain a cycle of length $l$. The minimum number of edges of a $C_l$-saturated graph was shown by Barefoot et al. to be between $n+c_1{n\over l}$ and $n+c_2{n\over l}$ for some positive constants $c_1$ and $c_2$. This confirmed a conjecture of Bollobás. Here we improve the value of $c_2$ for $l \geq 8$.

scholarly journals Saturation Numbers of Books

10.37236/842 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Guantao Chen ◽  
Ralph J. Faudree ◽  
Ronald J. Gould
Keyword(s):  
Saturated Graphs ◽  
Minimum Number ◽  
Saturation Number

A book $B_p$ is a union of $p$ triangles sharing one edge. This idea was extended to a generalized book $B_{b,p}$, which is the union of $p$ copies of a $K_{b+1}$ sharing a common $K_b$. A graph $G$ is called an $H$-saturated graph if $G$ does not contain $H$ as a subgraph, but $G\cup \{xy\}$ contains a copy of $H$, for any two nonadjacent vertices $x$ and $y$. The saturation number of $H$, denoted by $sat(H,n)$, is the minimum number of edges in $G$ for all $H$-saturated graphs $G$ of order $n$. We show that $$ sat(B_p, n) = {1\over2} \big( (p+1)(n-1) - \big\lceil {p\over2}\big\rceil \big\lfloor {p\over2} \big\rfloor + \theta(n,p)\big), $$ where $\theta(n, p) = \begin{cases} 1& \text{ if } p\equiv n -p/2 \equiv 0 \bmod 2 \\ 0& \text{ otherwise}\end{cases}$, provided $n \ge p^3 + p$. Moreover, we show that $$\eqalign{ sat(B_{b,p}, n) = \ & {1\over2} \big( (p+2b-3)(n-b+1) - \big\lceil {p\over2}\big\rceil \big\lfloor {p\over2} \big\rfloor\cr &+ \theta(n,p, b)+(b-1)(b-2) \big),\cr} $$ where $\theta(n, p, b) = \begin{cases} 1& \text{ if } p \equiv n -p/2 -b \equiv 0 \bmod 2 \\ 0 & \text{ otherwise} \end{cases}$, provided $n \ge 4(p+2b)^{b}$.

scholarly journals Saturated Graphs of Prescribed Minimum Degree

2016 ◽  
Vol 26 (2) ◽  
pp. 201-207 ◽  
Author(s):  
A. NICHOLAS DAY
Keyword(s):  
Minimum Degree ◽  
Upper Bounds ◽  
Saturated Graphs ◽  
Minimum Number

A graph G is H-saturated if it contains no copy of H as a subgraph but the addition of any new edge to G creates a copy of H. In this paper we are interested in the function satt(n,p), defined to be the minimum number of edges that a Kp-saturated graph on n vertices can have if it has minimum degree at least t. We prove that satt(n,p) = tn − O(1), where the limit is taken as n tends to infinity. This confirms a conjecture of Bollobás when p = 3. We also present constructions for graphs that give new upper bounds for satt(n,p).

scholarly journals On the Size of $(K_t,\mathcal{T}_k)$-Co-Critical Graphs

10.37236/8857 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Zi-Xia Song ◽  
Jingmei Zhang
Keyword(s):  
Graph Theory ◽  
Complete Graph ◽  
Minimum Degree ◽  
Bootstrap Percolation ◽  
Critical Graphs ◽  
Critical Graph ◽  
The Family ◽  
Saturated Graphs ◽  
Minimum Number ◽  
Monochromatic Copy

Given an integer $r\geqslant 1$ and graphs $G, H_1, \ldots, H_r$, we write $G \rightarrow ({H}_1, \ldots, {H}_r)$ if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$ for some $i\in\{1, \ldots, r\}$. A non-complete graph $G$ is $(H_1, \ldots, H_r)$-co-critical if $G \nrightarrow ({H}_1, \ldots, {H}_r)$, but $G+e\rightarrow ({H}_1, \ldots, {H}_r)$ for every edge $e$ in $\overline{G}$. In this paper, motivated by Hanson and Toft's conjecture [Edge-colored saturated graphs, J Graph Theory 11(1987), 191–196], we study the minimum number of edges over all $(K_t, \mathcal{T}_k)$-co-critical graphs on $n$ vertices, where $\mathcal{T}_k$ denotes the family of all trees on $k$ vertices. Following Day [Saturated graphs of prescribed minimum degree, Combin. Probab. Comput. 26 (2017), 201–207], we apply graph bootstrap percolation on a not necessarily $K_t$-saturated graph to prove that for all $t\geqslant4 $ and $k\geqslant\max\{6, t\}$, there exists a constant $c(t, k)$ such that, for all $n \ge (t-1)(k-1)+1$, if $G$ is a $(K_t, \mathcal{T}_k)$-co-critical graph on $n$ vertices, then $$ e(G)\geqslant \left(\frac{4t-9}{2}+\frac{1}{2}\left\lceil \frac{k}{2} \right\rceil\right)n-c(t, k).$$ Furthermore, this linear bound is asymptotically best possible when $t\in\{4,5\}$ and $k\geqslant6$. The method we develop in this paper may shed some light on attacking Hanson and Toft's conjecture.

Information capacity and bandwidth limitations in the SEM

1989 ◽  
Vol 47 ◽  
pp. 84-85
Author(s):  
D. C. Joy ◽  
R. D. Bunn
Keyword(s):  
Signal To Noise Ratio ◽  
Critical Dimension ◽  
Information Capacity ◽  
Acquisition Time ◽  
Signal To Noise ◽  
Sem Image ◽  
Probe Size ◽  
Minimum Number ◽  
Noise Ratio ◽  
Signal Channel

The information available from an SEM image is limited both by the inherent signal to noise ratio that characterizes the image and as a result of the transformations that it may undergo as it is passed through the amplifying circuits of the instrument. In applications such as Critical Dimension Metrology it is necessary to be able to quantify these limitations in order to be able to assess the likely precision of any measurement made with the microscope.The information capacity of an SEM signal, defined as the minimum number of bits needed to encode the output signal, depends on the signal to noise ratio of the image - which in turn depends on the probe size and source brightness and acquisition time per pixel - and on the efficiency of the specimen in producing the signal that is being observed. A detailed analysis of the secondary electron case shows that the information capacity C (bits/pixel) of the SEM signal channel could be written as :

Applying Generalizability Theory to Optimize Analysis of Spontaneous Teacher Talk in Elementary Classrooms

2020 ◽  
Vol 63 (6) ◽  
pp. 1947-1957
Author(s):  
Alexandra Hollo ◽  
Johanna L. Staubitz ◽  
Jason C. Chow
Keyword(s):  
Special Education Teachers ◽  
Generalizability Theory ◽  
Child Outcomes ◽  
Elementary Classrooms ◽  
Teacher Talk ◽  
Group Instruction ◽  
Data Set ◽  
School Year ◽  
Minimum Number ◽  
Representative Samples

Purpose Although sampling teachers' child-directed speech in school settings is needed to understand the influence of linguistic input on child outcomes, empirical guidance for measurement procedures needed to obtain representative samples is lacking. To optimize resources needed to transcribe, code, and analyze classroom samples, this exploratory study assessed the minimum number and duration of samples needed for a reliable analysis of conventional and researcher-developed measures of teacher talk in elementary classrooms. Method This study applied fully crossed, Person (teacher) × Session (samples obtained on 3 separate occasions) generalizability studies to analyze an extant data set of three 10-min language samples provided by 28 general and special education teachers recorded during large-group instruction across the school year. Subsequently, a series of decision studies estimated of the number and duration of sessions needed to obtain the criterion g coefficient ( g > .70). Results The most stable variables were total number of words and mazes, requiring only a single 10-min sample, two 6-min samples, or three 3-min samples to reach criterion. No measured variables related to content or complexity were adequately stable regardless of number and duration of samples. Conclusions Generalizability studies confirmed that a large proportion of variance was attributable to individuals rather than the sampling occasion when analyzing the amount and fluency of spontaneous teacher talk. In general, conventionally reported outcomes were more stable than researcher-developed codes, which suggests some categories of teacher talk are more context dependent than others and thus require more intensive data collection to measure reliably.

RGB Frequency Based Image Steganography

2020 ◽  
pp. 549-553
Author(s):  
Himanshu Kumar ◽  
Nitesh Kumar
Keyword(s):  
Image Steganography ◽  
Minimum Number

In this paper, we introduced a new RGB technique for image steganography. In this technique we introduced the idea of storing a different number of bits per channel (R, G or B) of a pixel based on the frequency of color values of pixel. The higher color frequency retains the maximum number of bits and lower color frequency stores the minimum number of bits.

On Number of Planes of Rearrangeably Nonblocking Optical Banyan Networks with Link Failures

2008 ◽  
Vol 1 (1) ◽  
pp. 43-54
Author(s):  
Basra Sultana ◽  
Mamun-ur-Rashid Khandker
Keyword(s):  
Optical Switching ◽  
Blocking Probability ◽  
Link Failure ◽  
Link Failures ◽  
Small Depth ◽  
Internal Link ◽  
Banyan Network ◽  
Minimum Number ◽  
Switch Design ◽  
Banyan Networks

Vertically stacked optical banyan (VSOB) networks are attractive for serving as optical switching systems due to the desirable properties (such as the small depth and self-routing capability) of banyan network structures. Although banyan-type networks result in severe blocking and crosstalk, both these problems can be minimized by using sufficient number of banyan planes in the VSOB network structure. The number of banyan planes is minimum for rearrangeably nonblocking and maximum for strictly nonblocking structure. Both results are available for VSOB networks when there exist no internal link-failures. Since the issue of link-failure is unavoidable, we intend to find the minimum number of planes required to make a VSOB network nonblocking when some links are broken or failed in the structure. This paper presents the approximate number of planes required to make a VSOB networks rearrangeably nonblocking allowing link-failures. We also show an interesting behavior of the  blocking  probability of a faulty VSOB networks that the blocking probability may not  always  increase monotonously with  the  increase  of  link-failures; blocking probability  decreases  for  certain range of  link-failures, and then increases again. We believe that such fluctuating behavior of blocking probability with the increase of link failure probability deserves special attention in switch design.  Keywords: Banyan networks; Blocking probability; Switching networks; Vertical stacking; Link-failures. © 2009 JSR Publications. ISSN: 2070-0237(Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v1i1.1070

Mathematics of a Sudo-Kurve

2018 ◽  
Vol 5 (10) ◽  
pp. 5-28
Author(s):  
Tanya Khovanova ◽  
Wayne Zhao
Keyword(s):  
Minimum Number ◽  
Distinct Solution ◽  
Equivalent Variant

Abstract We investigate a type of a Sudoku variant called Sudo-Kurve, which allows bent rows and columns, and develop a new, yet equivalent, variant we call a Sudo-Cube. We examine the total number of distinct solution grids for this type with or without symmetry. We study other mathematical aspects of this puzzle along with the minimum number of clues needed and the number of ways to place individual symbols.

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