scholarly journals On size multipartite Ramsey numbers for stars

2020 ◽  
Vol 3 (2) ◽  
pp. 109
Author(s):  
Anie Lusiani ◽  
Edy Tri Baskoro ◽  
Suhadi Wido Saputro
Keyword(s):  
Lower Bound ◽  
Disjoint Union ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ramsey Number ◽  
Complete Graphs ◽  
Ramsey Numbers ◽  
Simple Graphs ◽  
Multipartite Graphs ◽  
Necessary And Sufficient

<p>Burger and Vuuren defined the size multipartite Ramsey number for a pair of complete, balanced, multipartite graphs <em>mj</em>(<em>Ka</em>x<em>b</em>,<em>Kc</em>x<em>d</em>), for natural numbers <em>a,b,c,d</em> and <em>j</em>, where <em>a,c</em> &gt;= 2, in 2004. They have also determined the necessary and sufficient conditions for the existence of size multipartite Ramsey numbers <em>mj</em>(<em>Ka</em>x<em>b</em>,<em>Kc</em>x<em>d</em>). Syafrizal <em>et al</em>. generalized this definition by removing the completeness requirement. For simple graphs <em>G</em> and <em>H</em>, they defined the size multipartite Ramsey number <em>mj</em>(<em>G,H</em>) as the smallest natural number <em>t</em> such that any red-blue coloring on the edges of <em>Kj</em>x<em>t</em> contains a red <em>G</em> or a blue <em>H</em> as a subgraph. In this paper, we determine the necessary and sufficient conditions for the existence of multipartite Ramsey numbers <em>mj</em>(<em>G,H</em>), where both <em>G</em> and <em>H</em> are non complete graphs. Furthermore, we determine the exact values of the size multipartite Ramsey numbers <em>mj</em>(<em>K</em>1,<em>m</em>, <em>K</em>1,<em>n</em>) for all integers <em>m,n &gt;= </em>1 and <em>j </em>= 2,3, where <em>K</em>1,<em>m</em> is a star of order <em>m</em>+1. In addition, we also determine the lower bound of <em>m</em>3(<em>kK</em>1,<em>m</em>, <em>C</em>3), where <em>kK</em>1,<em>m</em> is a disjoint union of <em>k</em> copies of a star <em>K</em>1,<em>m</em> and <em>C</em>3 is a cycle of order 3.</p>

scholarly journals A strict upper bound for size multipartite Ramsey numbers of paths versus stars

2017 ◽  
Vol 1 (2) ◽  
pp. 9
Author(s):  
Chula Jayawardene
Keyword(s):  
Natural Number ◽  
Complete Graph ◽  
Upper Bound ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Infinite Class ◽  
Ramsey Numbers ◽  
Necessary And Sufficient

<p>Let $P_n$ represent the path of size $n$. Let $K_{1,m-1}$ represent a star of size $m$ and be denoted by $S_{m}$. Given a two coloring of the edges of a complete graph $K_{j \times s}$ we say that $K_{j \times s}\rightarrow (P_n,S_{m+1})$ if there is a copy of $P_n$ in the first color or a copy of $S_{m+1}$ in the second color. The size Ramsey multipartite number $m_j(P_n, S_{m+1})$ is the smallest natural number $s$ such that $K_{j \times s}\rightarrow (P_n,S_{m+1})$. Given $j,n,m$ if $s=\left\lceil \dfrac{n+m-1-k}{j-1} \right\rceil$, in this paper, we show that the size Ramsey numbers $m_j(P_n,S_{m+1})$ is bounded above by $s$ for $k=\left\lceil \dfrac{n-1}{j} \right\rceil$. Given $j\ge 3$ and $s$, we will obtain an infinite class $(n,m)$ that achieves this upper bound $s$. In the later part of the paper, will also investigate necessary and sufficient conditions needed for the upper bound to hold.</p>

On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences

2021 ◽  
Vol 71 (6) ◽  
pp. 1375-1400
Author(s):  
Feyzi Başar ◽  
Hadi Roopaei
Keyword(s):  
Lower Bound ◽  
Sequence Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Transformations ◽  
Infinite Matrix ◽  
The Matrix ◽  
Alpha Beta ◽  
Necessary And Sufficient ◽  
Bounded Sequences

Abstract Let F denote the factorable matrix and X ∈ {ℓp , c 0, c, ℓ ∞}. In this study, we introduce the domains X(F) of the factorable matrix in the spaces X. Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces X(F). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes (ℓ p (F), ℓ ∞), (ℓ p (F), f) and (X, Y(F)) of matrix transformations, where Y denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix F and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix F. Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix.

A Network-Flow Feasibility Theorem and Combinatorial Applications

1959 ◽  
Vol 11 ◽  
pp. 440-451 ◽  
Author(s):  
D. R. Fulkerson
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Network Flow ◽  
Present Author ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Private Communication ◽  
Circulatory Flow ◽  
Necessary And Sufficient ◽  
Circulation Theorem

There are a number of interesting theorems, relative to capacitated networks, that give necessary and sufficient conditions for the existence of flows satisfying constraints of various kinds. Typical of these are the supply-demand theorem due to Gale (4), which states a condition for the existence of a flow satisfying demands at certain nodes from supplies at other nodes, and the Hoffman circulation theorem (received by the present author in private communication), which states a condition for the existence of a circulatory flow in a network in which each arc has associated with it not only an upper bound for the arc flow, but a lower bound as well. If the constraints on flows are integral (for example, if the bounds on arc flows for the circulation theorem are integers), it is also true that integral flows meeting the requirements exist provided any flow does so.

scholarly journals Packings and Coverings of Various Complete Graphs with the 4-Cycle with a Pendant Edge

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Brandon Coker ◽  
Gary D. Coker ◽  
Robert Gardner ◽  
Yan Xia
Keyword(s):  
Sufficient Conditions ◽  
Bipartite Graphs ◽  
Necessary And Sufficient Conditions ◽  
Complete Graphs ◽  
Complete Bipartite Graphs ◽  
Necessary And Sufficient

We consider the packings and coverings of complete graphs with isomorphic copies of the 4-cycle with a pendant edge. Necessary and sufficient conditions are given for such structures for (1) complete graphs , (2) complete bipartite graphs , and (3) complete graphs with a hole . In the last two cases, we address both restricted and unrestricted coverings.

scholarly journals A Global Poincaré inequality on Graphs via a Conical Curvature-Dimension Condition

2018 ◽  
Vol 6 (1) ◽  
pp. 32-47 ◽  
Author(s):  
Sajjad Lakzian ◽  
Zachary Mcguirk
Keyword(s):  
Lower Bound ◽  
Sharp Estimate ◽  
Sufficient Conditions ◽  
Poincaré Inequality ◽  
Complete Graphs ◽  
Poincare Inequality ◽  
Underlying Graph ◽  
Dimension Analysis ◽  
Necessary And Sufficient ◽  
Sharp Lower Bound

Abstract We introduce and study the conical curvature-dimension condition, CCD(K, N), for finite graphs.We show that CCD(K, N) provides necessary and sufficient conditions for the underlying graph to satisfy a sharp global Poincaré inequality which in turn translates to a sharp lower bound for the first eigenvalues of these graphs. Another application of the conical curvature-dimension analysis is finding a sharp estimate on the curvature of complete graphs

scholarly journals On γ-Sets in Rings

2021 ◽  
Vol 14 (1) ◽  
pp. 314-326
Author(s):  
Eva Jenny C. Sigasig ◽  
Cristoper John S. Rosero ◽  
Michael Jr. Patula Baldado
Keyword(s):  
Lower Bound ◽  
Division Ring ◽  
Upper Bound ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Minimum Cardinality ◽  
Division Rings ◽  
Necessary And Sufficient

Let R be a ring with identity 1R. A subset J of R is called a γ-set if for every a ∈ R\J,there exist b, c ∈ J such that a+b = 0 and ac = 1R = ca. A γ-set of minimum cardinality is called a minimum γ-set. In this study, we identified some elements of R that are necessarily in a γ-sets, and we presented a method of constructing a new γ-set. Moreover, we gave: necessary and sufficient conditions for rings to have a unique γ-set; an upper bound for the total number of minimum γ-sets in a division ring; a lower bound for the total number of minimum γ-sets in a division ring; necessary and sufficient conditions for T(x) and T to be equal; necessary and sufficient conditions for a ring to have a trivial γ-set; necessary and sufficient conditions for an image of a γ-set to be a γ-set also; necessary and sufficient conditions for a ring to have a trivial γ-set; and, necessary and sufficient conditions for the families of γ-sets of two division rings to be isomorphic.

On the Borel-Cantelli Problem

1970 ◽  
Vol 13 (2) ◽  
pp. 273-275 ◽  
Author(s):  
Jonathan Shuster
Keyword(s):  
Lower Bound ◽  
Probability Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Let (Ω, , P) be a probability space, and A1, A2… be a sequence of members of . The classical Borel-Cantelli problem is to determine the probability that infinitely many events Ak occur. The classical results may be found in Feller [2, p. 188]; while related work may be found in Spitzer [3, p. 317], and Dawson and Sankoff [1]. The latter works are generalizations of the Borel-Cantelli lemmas, taken in different directions.In this paper, necessary and sufficient conditions will be given for infinitely many events Ak to occur, with probability 1. A lower bound for the probability that only finitely many Ak occur, is developed. In addition, necessary and sufficient conditions that only finitely many Ak occur, with probability 1, are given.

Amalgamated Factorizations of Complete Graphs

1994 ◽  
Vol 3 (2) ◽  
pp. 215-232 ◽  
Author(s):  
J. K. Dugdale ◽  
A. J. W. Hilton
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complete Graphs ◽  
Necessary And Sufficient

We give some sufficient conditions for an (S, U)-outline T-factorization of Kn to be an (S, U)-amalgamated T-factorization of Kn. We then apply these to give various necessary and sufficient conditions for edge coloured graphs G to have recoverable embeddings in T-factorized Kn's.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj&gt; 0 for eachj&gt; 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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