scholarly journals New Computational Upper Bounds for Ramsey Numbers $R(3,k)$

10.37236/2824 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Jan Goedgebeur ◽  
Stanisław P. Radziszowski
Keyword(s):  
Lower Bounds ◽  
Independent Sets ◽  
Upper Bounds ◽  
Computational Techniques ◽  
Free Graph ◽  
Ramsey Numbers ◽  
Critical Graph ◽  
Minimum Number

Using computational techniques we derive six new upper bounds on the classical two-color Ramsey numbers: $R(3,10) \le 42$, $R(3,11) \le 50$, $R(3,13) \le 68$, $R(3,14) \le 77$, $R(3,15) \le 87$, and $R(3,16) \le 98$. All of them are improvements by one over the previously best known bounds. Let $e(3,k,n)$ denote the minimum number of edges in any triangle-free graph on $n$ vertices without independent sets of order $k$. The new upper bounds on $R(3,k)$ are obtained by completing the computation of the exact values of $e(3,k,n)$ for all $n$ with $k \leq 9$ and for all $n \leq 33$ for $k = 10$, and by establishing new lower bounds on $e(3,k,n)$ for most of the open cases for $10 \le k \le 15$. The enumeration of all graphs witnessing the values of $e(3,k,n)$ is completed for all cases with $k \le 9$. We prove that the known critical graph for $R(3,9)$ on 35 vertices is unique up to isomorphism. For the case of $R(3,10)$, first we establish that $R(3,10)=43$ if and only if $e(3,10,42)=189$, or equivalently, that if $R(3,10)=43$ then every critical graph is regular of degree 9. Then, using computations, we disprove the existence of the latter, and thus show that $R(3,10) \le 42$.

scholarly journals The Ramsey Number $R(3,K_{10}-e)$ and Computational Bounds for $R(3,G)$

10.37236/3684 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Jan Goedgebeur ◽  
Stanisław P. Radziszowski
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Ramsey Number ◽  
Computer Algorithms ◽  
Free Graph ◽  
Ramsey Numbers ◽  
Open Case ◽  
Minimum Number ◽  
Graph Generation

Using computer algorithms we establish that the Ramsey number $R(3,K_{10}-e)$ is equal to 37, which solves the smallest open case for Ramsey numbers of this type. We also obtain new upper bounds for the cases of $R(3,K_k-e)$ for $11 \le k \le 16$, and show by construction a new lower bound $55 \le R(3,K_{13}-e)$.The new upper bounds on $R(3,K_k-e)$ are obtained by using the values and lower bounds on $e(3,K_l-e,n)$ for $l \le k$, where $e(3,K_k-e,n)$ is the minimum number of edges in any triangle-free graph on $n$ vertices without $K_k-e$ in the complement. We complete the computation of the exact values of $e(3,K_k-e,n)$ for all $n$ with $k \leq 10$ and for $n \leq 34$ with $k = 11$, and establish many new lower bounds on $e(3,K_k-e,n)$ for higher values of $k$.Using the maximum triangle-free graph generation method, we determine two other previously unknown Ramsey numbers, namely $R(3,K_{10}-K_3-e)=31$ and $R(3,K_{10}-P_3-e)=31$. For graphs $G$ on 10 vertices, besides $G=K_{10}$, this leaves 6 open cases of the form $R(3,G)$. The hardest among them appears to be $G=K_{10}-2K_2$, for which we establish the bounds $31 \le R(3,K_{10}-2K_2) \le 33$.

scholarly journals Lower Bounds for Maximal Matchings and Maximal Independent Sets

2021 ◽  
Vol 68 (5) ◽  
pp. 1-30
Author(s):  
Alkida Balliu ◽  
Sebastian Brandt ◽  
Juho Hirvonen ◽  
Dennis Olivetti ◽  
Mikaël Rabie ◽  
...  
Keyword(s):  
Lower Bounds ◽  
Independent Set ◽  
Independent Sets ◽  
Deterministic Algorithm ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Maximal Independent Set ◽  
Maximal Matching ◽  
Maximal Independent Sets ◽  
Open Question

There are distributed graph algorithms for finding maximal matchings and maximal independent sets in O ( Δ + log * n ) communication rounds; here, n is the number of nodes and Δ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the dependency on n is optimal: These problems cannot be solved in o (log * n ) rounds even if Δ = 2. However, the dependency on Δ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds. We prove that the upper bounds are tight. We show that any algorithm that finds a maximal matching or maximal independent set with probability at least 1-1/ n requires Ω (min { Δ , log log n / log log log n }) rounds in the LOCAL model of distributed computing. As a corollary, it follows that any deterministic algorithm that finds a maximal matching or maximal independent set requires Ω (min { Δ , log n / log log n }) rounds; this is an improvement over prior lower bounds also as a function of  n .

scholarly journals UNKNOTTING NUMBERS AND TRIPLE POINT CANCELLING NUMBERS OF TORUS-COVERING KNOTS

2013 ◽  
Vol 22 (03) ◽  
pp. 1350010
Author(s):  
INASA NAKAMURA
Keyword(s):  
Finite Number ◽  
Lower Bounds ◽  
Triple Point ◽  
Upper Bounds ◽  
Minimum Number ◽  
Triple Points

It is known that any surface knot can be transformed to an unknotted surface knot or a surface knot which has a diagram with no triple points by a finite number of 1-handle additions. The minimum number of such 1-handles is called the unknotting number or the triple point cancelling number, respectively. In this paper, we give upper bounds and lower bounds of unknotting numbers and triple point cancelling numbers of torus-covering knots, which are surface knots in the form of coverings over the standard torus T. Upper bounds are given by using m-charts on T presenting torus-covering knots, and lower bounds are given by using quandle colorings and quandle cocycle invariants.

scholarly journals Fractional Matchings, Component-Factors and Edge-Chromatic Critical Graphs

2021 ◽  
Author(s):  
Antje Klopp ◽  
Eckhard Steffen
Keyword(s):  
Upper Bounds ◽  
Matching Number ◽  
Critical Graphs ◽  
Critical Graph ◽  
Minimum Number

AbstractThe first part of the paper studies star-cycle factors of graphs. It characterizes star-cycle factors of a graph G and proves upper bounds for the minimum number of $$K_{1,2}$$ K 1 , 2 -components in a $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$ { K 1 , 1 , K 1 , 2 , C n : n ≥ 3 } -factor of a graph G. Furthermore, it shows where these components are located with respect to the Gallai–Edmonds decomposition of G and it characterizes the edges which are not contained in any $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$ { K 1 , 1 , K 1 , 2 , C n : n ≥ 3 } -factor of G. The second part of the paper proves that every edge-chromatic critical graph G has a $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$ { K 1 , 1 , K 1 , 2 , C n : n ≥ 3 } -factor, and the number of $$K_{1,2}$$ K 1 , 2 -components is bounded in terms of its fractional matching number. Furthermore, it shows that for every edge e of G, there is a $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$ { K 1 , 1 , K 1 , 2 , C n : n ≥ 3 } -factor F with $$e \in E(F)$$ e ∈ E ( F ) . Consequences of these results for Vizing’s critical graph conjectures are discussed.

scholarly journals Counting Independent Sets in Hypergraphs

2014 ◽  
Vol 23 (4) ◽  
pp. 539-550 ◽  
Author(s):  
JEFF COOPER ◽  
KUNAL DUTTA ◽  
DHRUV MUBAYI
Keyword(s):  
Recent Result ◽  
Lower Bounds ◽  
Independent Set ◽  
Independent Sets ◽  
Average Degree ◽  
General Statement ◽  
Free Graph ◽  
Hereditary Properties ◽  
Image Position ◽  
Linear Hypergraph

Let G be a triangle-free graph with n vertices and average degree t. We show that G contains at least ${\exp\biggl({1-n^{-1/12})\frac{1}{2}\frac{n}{t}\ln t} \biggl(\frac{1}{2}\ln t-1\biggr)\biggr)}$ independent sets. This improves a recent result of the first and third authors [8]. In particular, it implies that as n → ∞, every triangle-free graph on n vertices has at least ${e^{(c_1-o(1)) \sqrt{n} \ln n}}$ independent sets, where $c_1 = \sqrt{\ln 2}/4 = 0.208138 \ldots$. Further, we show that for all n, there exists a triangle-free graph with n vertices which has at most ${e^{(c_2+o(1))\sqrt{n}\ln n}}$ independent sets, where $c_2 = 2\sqrt{\ln 2} = 1.665109 \ldots$. This disproves a conjecture from [8].Let H be a (k+1)-uniform linear hypergraph with n vertices and average degree t. We also show that there exists a constant ck such that the number of independent sets in H is at least ${\exp\biggl({c_{k} \frac{n}{t^{1/k}}\ln^{1+1/k}{t}\biggr})}.$ This is tight apart from the constant ck and generalizes a result of Duke, Lefmann and Rödl [9], which guarantees the existence of an independent set of size $\Omega\biggl(\frac{n}{t^{1/k}} \ln^{1/k}t\biggr).$ Both of our lower bounds follow from a more general statement, which applies to hereditary properties of hypergraphs.

On the Size of the Wild Set

2005 ◽  
Vol 57 (1) ◽  
pp. 180-203
Author(s):  
Marius Somodi
Keyword(s):  
Lower Bounds ◽  
Algebraic Number ◽  
Number Fields ◽  
Exact Formula ◽  
Upper Bounds ◽  
Algebraic Number Fields ◽  
Witt Rings ◽  
Minimum Number

AbstractTo every pair of algebraic number fields with isomorphic Witt rings one can associate a number, called the minimum number of wild primes. Earlier investigations have established lower bounds for this number. In this paper an analysis is presented that expresses the minimum number of wild primes in terms of the number of wild dyadic primes. This formula not only gives immediate upper bounds, but can be considered to be an exact formula for the minimum number of wild primes.

scholarly journals On-line Ramsey Numbers of Paths and Cycles

10.37236/4097 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Joanna Cyman ◽  
Tomasz Dzido ◽  
John Lapinskas ◽  
Allan Lo
Keyword(s):  
Lower Bounds ◽  
Ramsey Number ◽  
Additive Constant ◽  
Ramsey Numbers ◽  
Minimum Number ◽  
On Line

Consider a game played on the edge set of the infinite clique by two players, Builder and Painter. In each round, Builder chooses an edge and Painter colours it red or blue. Builder wins by creating either a red copy of $G$ or a blue copy of $H$ for some fixed graphs $G$ and $H$. The minimum number of rounds within which Builder can win, assuming both players play perfectly, is the on-line Ramsey number $\tilde{r}(G,H)$. In this paper, we consider the case where $G$ is a path $P_k$. We prove that $\tilde{r}(P_3,P_{\ell+1}) = \lceil 5\ell/4\rceil = \tilde{r}(P_3,C_{\ell})$ for all $\ell \ge 5$, and determine $\tilde{r}(P_4,P_{\ell+1})$ up to an additive constant for all $\ell \ge 3$. We also prove some general lower bounds for on-line Ramsey numbers of the form $\tilde{r}(P_{k+1},H)$.

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Ramsey Numbers

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.

scholarly journals A Note on On-Line Ramsey Numbers for Some Paths

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 735
Author(s):  
Tomasz Dzido ◽  
Renata Zakrzewska
Keyword(s):  
Ramsey Number ◽  
Ramsey Numbers ◽  
Minimum Number ◽  
On Line ◽  
Infinite Set ◽  
Do So

We consider the important generalisation of Ramsey numbers, namely on-line Ramsey numbers. It is easiest to understand them by considering a game between two players, a Builder and Painter, on an infinite set of vertices. In each round, the Builder joins two non-adjacent vertices with an edge, and the Painter colors the edge red or blue. An on-line Ramsey number r˜(G,H) is the minimum number of rounds it takes the Builder to force the Painter to create a red copy of graph G or a blue copy of graph H, assuming that both the Builder and Painter play perfectly. The Painter’s goal is to resist to do so for as long as possible. In this paper, we consider the case where G is a path P4 and H is a path P10 or P11.

scholarly journals Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations

2020 ◽  
Vol 26 (2) ◽  
pp. 131-161
Author(s):  
Florian Bourgey ◽  
Stefano De Marco ◽  
Emmanuel Gobet ◽  
Alexandre Zhou
Keyword(s):  
Monte Carlo ◽  
Lower Bounds ◽  
Asymptotic Optimality ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Independent Random Variables ◽  
Outer Function ◽  
Control Problems ◽  
Multilevel Monte Carlo ◽  
Primal Dual

AbstractThe multilevel Monte Carlo (MLMC) method developed by M. B. Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56 2008, 3, 607–617] has a natural application to the evaluation of nested expectations {\mathbb{E}[g(\mathbb{E}[f(X,Y)|X])]}, where {f,g} are functions and {(X,Y)} a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of initial margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotic optimality; at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal-dual algorithms for stochastic control problems.

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