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 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 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 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 Ramsey Numbers of Ordered Graphs

10.37236/7816 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Martin Balko ◽  
Josef Cibulka ◽  
Karel Král ◽  
Jan Kynčl
Keyword(s):  
Chromatic Number ◽  
Complete Graph ◽  
Positive Answer ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Minimum Number ◽  
Ordered Graphs ◽  
Constant Interval ◽  
Monochromatic Copy ◽  
Special Classes

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by two colors contains a monochromatic copy of $\mathcal{G}$. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which $\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number $\overline{R}(\mathcal{G})$ is polynomial in the number of vertices of $\mathcal{G}$ if the bandwidth of $\mathcal{G}$ is constant or if $\mathcal{G}$ is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by Károlyi, Pach, Tóth, and Valtr.

scholarly journals Generalized Ramsey numbers for paths in2-chromatic graphs

1986 ◽  
Vol 9 (2) ◽  
pp. 273-276 ◽  
Author(s):  
R. Meenakshi ◽  
P. S. Sundararaghavan
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Colored Graph ◽  
Ramsey Numbers

Chung and Liu have defined thed-chromatic Ramsey number as follows. Let1≤d≤cand lett=(cd). Let1,2,…,tbe the ordered subsets ofdcolors chosen fromcdistinct colors. LetG1,G2,…,Gtbe graphs. Thed-chromatic Ramsey number denoted byrdc(G1,G2,…,Gt)is defined as the least numberpsuch that, if the edges of the complete graphKpare colored in any fashion withccolors, then for somei, the subgraph whose edges are colored in theith subset of colors contains aGi. In this paper it is shown thatr23(Pi,Pj,Pk)=[(4k+2j+i−2)/6]wherei≤j≤k<r(Pi,Pj),r23stands for a generalized Ramsey number on a2-colored graph andPiis a path of orderi.

scholarly journals Avoidance colourings for small nonclassical Ramsey numbers

2011 ◽  
Vol Vol. 13 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Alewyn Petrus Burger ◽  
Jan H. Vuuren
Keyword(s):  
Natural Number ◽  
Complete Graph ◽  
Domination Number ◽  
Ramsey Number ◽  
Computer Search ◽  
Ramsey Numbers ◽  
Blue Edge ◽  
International Audience ◽  
Upper Domination Number ◽  
Edge Colouring

Graphs and Algorithms International audience The irredundant Ramsey number s - s(m, n) [upper domination Ramsey number u - u(m, n), respectively] is the smallest natural number s [u, respectively] such that in any red-blue edge colouring (R, B) of the complete graph of order s [u, respectively], it holds that IR(B) \textgreater= m or IR(R) \textgreater= n [Gamma (B) \textgreater= m or Gamma(R) \textgreater= n, respectively], where Gamma and IR denote respectively the upper domination number and the irredundance number of a graph. Furthermore, the mixed irredundant Ramsey number t = t(m, n) [mixed domination Ramsey number v = v(m, n), respectively] is the smallest natural number t [v, respectively] such that in any red-blue edge colouring (R, B) of the complete graph of order t [v, respectively], it holds that IR(B) \textgreater= m or beta(R) \textgreater= n [Gamma(B) \textgreater= m or beta(R) \textgreater= n, respectively], where beta denotes the independent domination number of a graph. These four classes of non-classical Ramsey numbers have previously been studied in the literature. In this paper we introduce a new Ramsey number w = w(m, n), called the irredundant-domination Ramsey number, which is the smallest natural number w such that in any red-blue edge colouring (R, B) of the complete graph of order w, it holds that IR(B) \textgreater= m or Gamma(R) \textgreater= n. A computer search is employed to determine complete sets of avoidance colourings of small order for these five classes of nonclassical Ramsey numbers. In the process the fifteen previously unknown Ramsey numbers t(4, 4) = 14, t(6, 3) = 17, u(4, 4) = 13, v(4, 3) = 8, v(4, 4) = 14, v(5, 3) = 13, v(6, 3) = 17, w(3, 3) = 6, w(3, 4) = w(4, 3) = 8, w(4, 4) = 13, w(3, 5) = w(5, 3) = 12 and w(3, 6) = w(6, 3) = 15 are established.

scholarly journals On Some Ramsey and Turán-Type Numbers for Paths and Cycles

10.37236/1081 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Tomasz Dzido ◽  
Marek Kubale ◽  
Konrad Piwakowski
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Multicolor Ramsey Number ◽  
Monochromatic Copy

For given graphs $G_{1}, G_{2}, ... , G_{k}$, where $k \geq 2$, the multicolor Ramsey number $R(G_{1}, G_{2}, ... , G_{k})$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph on $n$ vertices with $k$ colors, there is always a monochromatic copy of $G_{i}$ colored with $i$, for some $1 \leq i \leq k$. Let $P_k$ (resp. $C_k$) be the path (resp. cycle) on $k$ vertices. In the paper we show that $R(P_3,C_k,C_k)=R(C_k,C_k)=2k-1$ for odd $k$. In addition, we provide the exact values for Ramsey numbers $R(P_{4}, P_{4}, C_{k})=k+2$ and $R(P_{3}, P_{5}, C_{k})=k+1$.

scholarly journals Three Color Ramsey Numbers for Graphs with at most 4 Vertices

10.37236/2160 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Luis Boza ◽  
Janusz Dybizbański ◽  
Tomasz Dzido
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Free Graph ◽  
Ramsey Numbers ◽  
Monochromatic Copy

For given graphs $H_{1}, H_{2}, H_{3}$, the 3-color Ramsey number $R(H_{1},$ $H_{2}, H_{3})$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph of order $n$ with $3$ colors, then it always contains a monochromatic copy of $H_{i}$ colored with $i$, for some $1 \leq i \leq 3$.We study the bounds on 3-color Ramsey numbers $R(H_1,H_2,H_3)$, where $H_i$ is an isolate-free graph different from $K_2$ with at most four vertices, establishing that $R(P_4,C_4,K_4)=14$, $R(C_4,K_3,K_4-e)=17$, $R(C_4,K_3+e,K_4-e)=17$, $R(C_4,K_4-e,K_4-e)=19$, $28\le R(C_4,K_4-e,K_4)\le36$, $R(K_3,K_4-e,K_4)\le41$, $R(K_4-e,K_4-e,K_4)\le59$ and $R(K_4-e,K_4,K_4)\le113$. Also, we prove that $R(K_3+e,K_4-e,K_4-e)=R(K_3,K_4-e,K_4-e)$, $R(C_4,K_3+e,K_4)\le\max\{R(C_4,K_3,K_4),29\}\le32$, $R(K_3+e,K_4-e,K_4)\le\max\{R(K_3,K_4-e,K_4),33\}\le41$ and $R(K_3+e,K_4,K_4)\le\max\{R(K_3,K_4,K_4),2R(K_3,K_3,K_4)+2\}\le79$.This paper is an extension of the article by Arste, Klamroth, Mengersen [Utilitas Mathematica, 1996].

On Ramsey Numbers of Sparse Graphs

2003 ◽  
Vol 12 (5-6) ◽  
pp. 627-641 ◽  
Author(s):  
Alexander Kostochka ◽  
B Sudakov
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Sparse Graphs ◽  
Minimum Integer ◽  
Image Position ◽  
Monochromatic Copy

The Ramsey number, , of a graph G is the minimum integer N such that, in every 2-colouring of the edges of the complete graph on N vertices, there is a monochromatic copy of G. In 1975, Burr and Erdős posed a problem on Ramsey numbers of d-degenerate graphs, i.e., graphs in which every subgraph has a vertex of degree at most d. They conjectured that for every d there exists a constant c(d) such that for any d-degenerate graph G of order n.In this paper we prove that for each such G. In fact, we show that, for every , sufficiently large n, and any graph H of order , either H or its complement contains a (d,n)-common graph, that is, a graph in which every set of d vertices has at least n common neighbours. It is easy to see that any (d,n)-common graph contains every d-degenerate graph G of order n. We further show that, for every constant C, there is an n and a graph H of order such that neither H nor its complement contains a -common graph.

scholarly journals New Lower Bound for Multicolor Ramsey Numbers for Even Cycles

10.37236/1980 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Tomasz Dzido ◽  
Andrzej Nowik ◽  
Piotr Szuca
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Finite Family ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Multicolor Ramsey Number ◽  
Monochromatic Copy

For given finite family of graphs $G_{1}, G_{2}, \ldots , G_{k}, k \geq 2$, the multicolor Ramsey number $R(G_{1}, G_{2}, \ldots , G_{k})$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph on $n$ vertices with $k$ colors then there is always a monochromatic copy of $G_{i}$ colored with $i$, for some $1 \leq i \leq k$. We give a lower bound for $k-$color Ramsey number $R(C_{m}, C_{m}, \ldots , C_{m})$, where $m \geq 4$ is even and $C_{m}$ is the cycle on $m$ vertices.

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