scholarly journals A Note on Odd Cycle-Complete Graph Ramsey Numbers

10.37236/1662 ◽  
2001 ◽  
Vol 9 (1) ◽  
Author(s):  
Benny Sudakov
Keyword(s):  
Positive Integer ◽  
Complete Graph ◽  
Upper Bound ◽  
Short Note ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Odd Cycle

The Ramsey number $r(C_l, K_n)$ is the smallest positive integer $m$ such that every graph of order $m$ contains either cycle of length $l$ or a set of $n$ independent vertices. In this short note we slightly improve the best known upper bound on $r(C_l, K_n)$ for odd $l$.

scholarly journals A Note on Restricted Online Ramsey Numbers of Matchings

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Vojtěch Dvořák
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Short Note ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Minimum Number

Consider the following game between Builder and Painter. We take some families of graphs $\mathcal{G}_{1},\ldots,\mathcal{G}_t$ and an integer $n$ such that $n \geq R(\mathcal{G}_1,\ldots,\mathcal{G}_t)$. In each turn, Builder picks an edge of initially uncoloured $K_n$ and Painter colours that edge with some colour $i \in \left\{ 1,\ldots,t \right\}$ of her choice. The game ends when a graph $G_i$ in colour $i $ for some $G_i \in \mathcal{G}_i$ and some $i$ is created. The restricted online Ramsey number $\tilde{R}(\mathcal{G}_{1},\ldots,\mathcal{G}_t;n)$ is the minimum number of turns that Builder needs to guarantee the game to end. In a recent paper, Briggs and Cox studied the restricted online Ramsey numbers of matchings and determined a general upper bound for them. They proved that for $n=3r-1=R_2(r K_2)$ we have $\tilde{R}_{2}(r K_2;n) \leq n-1$ and asked whether this was tight. In this short note, we provide a general lower bound for these Ramsey numbers. As a corollary, we answer this question of Briggs and Cox, and confirm that for $n=3r-1$ we have $\tilde{R}_{2}(r K_2;n) = n-1$. We also show that for $n'=4r-2=R_3(r K_2)$ we have $\tilde{R}_{3}(r K_2;n') = 5r-4$.

scholarly journals Ramsey Numbers for Theta Graphs

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
M. M. M. Jaradat ◽  
M. S. A. Bataineh ◽  
S. M. E. Radaideh
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Subgraph Isomorphic

The graph Ramsey number is the smallest integer with the property that any complete graph of at least vertices whose edges are colored with two colors (say, red and blue) contains either a subgraph isomorphic to all of whose edges are red or a subgraph isomorphic to all of whose edges are blue. In this paper, we consider the Ramsey numbers for theta graphs. We determine , for . More specifically, we establish that for . Furthermore, we determine for . In fact, we establish that if is even, if is odd.

scholarly journals The Size, Multipartite Ramsey Numbers for nK2 Versus Path–Path and Cycle

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 764
Author(s):  
Yaser Rowshan ◽  
Mostafa Gholami ◽  
Stanford Shateyi
Keyword(s):  
Positive Integer ◽  
Ramsey Number ◽  
Complete Multipartite Graph ◽  
Ramsey Numbers ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Monochromatic Copy

For given graphs G1,G2,…,Gn and any integer j, the size of the multipartite Ramsey number mj(G1,G2,…,Gn) is the smallest positive integer t such that any n-coloring of the edges of Kj×t contains a monochromatic copy of Gi in color i for some i, 1≤i≤n, where Kj×t denotes the complete multipartite graph having j classes with t vertices per each class. In this paper, we computed the size of the multipartite Ramsey numbers mj(K1,2,P4,nK2) for any j,n≥2 and mj(nK2,C7), for any j≤4 and n≥2.

scholarly journals Ramsey Numbers of Trees Versus Odd Cycles

10.37236/5731 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Matthew Brennan
Keyword(s):  
Positive Integer ◽  
Constant Factor ◽  
Linear Functions ◽  
Ramsey Numbers ◽  
Odd Cycle ◽  
Odd Cycles

Burr, Erdős, Faudree, Rousseau and Schelp initiated the study of Ramsey numbers of trees versus odd cycles, proving that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 756m^{10}$, where $T_n$ is a tree with $n$ vertices and $C_m$ is an odd cycle of length $m$. They proposed to study the minimum positive integer $n_0(m)$ such that this result holds for all $n \ge n_0(m)$, as a function of $m$. In this paper, we show that $n_0(m)$ is at most linear. In particular, we prove that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 25m$. Combining this with a result of Faudree, Lawrence, Parsons and Schelp yields $n_0(m)$ is bounded between two linear functions, thus identifying $n_0(m)$ up to a constant factor.

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>

scholarly journals Constrained Ramsey Numbers

2009 ◽  
Vol 18 (1-2) ◽  
pp. 247-258 ◽  
Author(s):  
PO-SHEN LOH ◽  
BENNY SUDAKOV
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Ramsey Theorem ◽  
Logarithmic Factor ◽  
Subgraph Isomorphic ◽  
Rate Of Growth ◽  
Special Cases ◽  
Edge Colouring

For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge colouring of the complete graph on n vertices (with any number of colours) has a monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T. Here, a subgraph is said to be rainbow if all of its edges have different colours. It is an immediate consequence of the Erdős–Rado Canonical Ramsey Theorem that f(S, T) exists if and only if S is a star or T is acyclic. Much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang and Ling showed that f(S, T) ≤ O(st2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this paper, we study this case and show that f(S, Pt) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.

scholarly journals A Note on Edge-Colourings Avoiding Rainbow $K_4$ and Monochromatic $K_m$

10.37236/257 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Veselin Jungić ◽  
Tomáš Kaiser ◽  
Daniel Král'
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Projective Planes ◽  
Ramsey Number ◽  
Complete Subgraph ◽  
Finite Projective Planes ◽  
Edge Colourings ◽  
Edge Colouring

We study the mixed Ramsey number $maxR(n,{K_m},{K_r})$, defined as the maximum number of colours in an edge-colouring of the complete graph $K_n$, such that $K_n$ has no monochromatic complete subgraph on $m$ vertices and no rainbow complete subgraph on $r$ vertices. Improving an upper bound of Axenovich and Iverson, we show that $maxR(n,{K_m},{K_4}) \leq n^{3/2}\sqrt{2m}$ for all $m\geq 3$. Further, we discuss a possible way to improve their lower bound on $maxR(n,{K_4},{K_4})$ based on incidence graphs of finite projective planes.

scholarly journals Un método algoritmo para el cálculo del número baricéntrico de Ramsey para el grafo estrella

2018 ◽  
Vol 36 (2) ◽  
pp. 169-183
Author(s):  
Felicia Villarroel ◽  
J. Figueroa ◽  
H. Márquez ◽  
A. Anselmi
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Complete Graph ◽  
Ramsey Number ◽  
Combinatorial Theory ◽  
Star Graph ◽  
Definition Of

Let G be an abelian finite group and H be a graph. A sequence in G, with length al least two, is barycentric if it contains an ”average” element of its terms. Within the context of these sequences, one defines the barycentric Ramsey number, denoted by BR(H, G), as the smallest positive integer t such that any coloration of the edges of the complete graph Kt with elements of G produces a barycentric copy of the graph H. In this work we present a method based on the combinatorial theory and on the definition of barycentric Ramsey for calculating exact values of the above metioned constant, for some small graphs where the order is less than or equal to 8. We will exemplify the case where H is the star graph K1,k, and where G is the cyclical group Zn, with 3 ≤ n ≤ 11 and 3 ≤ k ≤ n.

scholarly journals Ramsey Numbers of Connected Clique Matchings

10.37236/6284 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Barnaby Roberts
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Connected Subgraph ◽  
Ramsey Numbers

We determine the Ramsey number of a connected clique matching. That is, we show that if $G$ is a $2$-edge-coloured complete graph on $(r^2-r-1)n-r+1$ vertices, then there is a monochromatic connected subgraph containing $n$ disjoint copies of $K_r$, and that this number of vertices cannot be reduced.

scholarly journals Off-Diagonal Ordered Ramsey Numbers of Matchings

10.37236/8085 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Dhruv Rohatgi
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Edge Coloring ◽  
Ramsey Number ◽  
Relative Order ◽  
Ramsey Numbers ◽  
Asymptotic Bounds ◽  
Special Cases ◽  
Ordered Graphs

For ordered graphs $G$ and $H$, the ordered Ramsey number $r_<(G,H)$ is the smallest $n$ such that every red/blue edge coloring of the complete ordered graph on vertices $\{1,\dots,n\}$ contains either a blue copy of $G$ or a red copy of $H$, where the embedding must preserve the relative order of vertices. One number of interest, first studied by Conlon, Fox, Lee, and Sudakov, is the off-diagonal ordered Ramsey number $r_<(M, K_3)$, where $M$ is an ordered matching on $n$ vertices. In particular, Conlon et al. asked what asymptotic bounds (in $n$) can be obtained for $\max r_<(M, K_3)$, where the maximum is over all ordered matchings $M$ on $n$ vertices. The best-known upper bound is $O(n^2/\log n)$, whereas the best-known lower bound is $\Omega((n/\log n)^{4/3})$, and Conlon et al. hypothesize that there is some fixed $\epsilon > 0$ such that $r_<(M, K_3) = O(n^{2-\epsilon})$ for every ordered matching $M$. We resolve two special cases of this conjecture. We show that the off-diagonal ordered Ramsey numbers for ordered matchings in which edges do not cross are nearly linear. We also prove a truly sub-quadratic upper bound for random ordered matchings with interval chromatic number $2$.

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