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 Cycle-Complete Ramsey Numbers

2019 ◽  
Author(s):  
Peter Keevash ◽  
Eoin Long ◽  
Jozef Skokan
Keyword(s):  
Natural Number ◽  
Absolute Constant ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Blue Edge ◽  
Edge Colouring

Abstract The Ramsey number $r(C_{\ell },K_n)$ is the smallest natural number $N$ such that every red/blue edge colouring of a clique of order $N$ contains a red cycle of length $\ell $ or a blue clique of order $n$. In 1978, Erd̋s, Faudree, Rousseau, and Schelp conjectured that $r(C_{\ell },K_n) = (\ell -1)(n-1)+1$ for $\ell \geq n\geq 3$ provided $(\ell ,n) \neq (3,3)$. We prove that, for some absolute constant $C\ge 1$, we have $r(C_{\ell },K_n) = (\ell -1)(n-1)+1$ provided $\ell \geq C\frac{\log n}{\log \log n}$. Up to the value of $C$ this is tight since we also show that, for any $\varepsilon>0$ and $n> n_0(\varepsilon )$, we have $r(C_{\ell }, K_n) \gg (\ell -1)(n-1)+1$ for all $3 \leq \ell \leq (1-\varepsilon )\frac{\log n}{\log \log n}$. This proves the conjecture of Erd̋s, Faudree, Rousseau, and Schelp for large $\ell $, a stronger form of the conjecture due to Nikiforov, and answers (up to multiplicative constants) two further questions of Erd̋s, Faudree, Rousseau, and Schelp.

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 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 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 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 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 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 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 On the Size-Ramsey Number of Cycles

2019 ◽  
Vol 28 (06) ◽  
pp. 871-880
Author(s):  
R. Javadi ◽  
F. Khoeini ◽  
G. R. Omidi ◽  
A. Pokrovskiy
Keyword(s):  
Upper Bounds ◽  
Ramsey Number ◽  
Alternative Proof ◽  
Regularity Lemma ◽  
Ramsey Numbers ◽  
Szemeredi's Regularity Lemma ◽  
Number Of Cycles ◽  
Image Position ◽  
Edge Colouring ◽  
Monochromatic Copy

AbstractFor given graphs G1,…, Gk, the size-Ramsey number $\hat R({G_1}, \ldots ,{G_k})$ is the smallest integer m for which there exists a graph H on m edges such that in every k-edge colouring of H with colours 1,…,k, H contains a monochromatic copy of Gi of colour i for some 1 ≤ i ≤ k. We denote $\hat R({G_1}, \ldots ,{G_k})$ by ${\hat R_k}(G)$ when G1 = ⋯ = Gk = G.Haxell, Kohayakawa and Łuczak showed that the size-Ramsey number of a cycle Cn is linear in n, ${\hat R_k}({C_n}) \le {c_k}n$ for some constant ck. Their proof, however, is based on Szemerédi’s regularity lemma so no specific constant ck is known.In this paper, we give various upper bounds for the size-Ramsey numbers of cycles. We provide an alternative proof of ${\hat R_k}({C_n}) \le {c_k}n$ , avoiding use of the regularity lemma, where ck is exponential and doubly exponential in k, when n is even and odd, respectively. In particular, we show that for sufficiently large n we have ${\hat R_2}({C_n}) \le {10^5} \times cn$ , where c = 6.5 if n is even and c = 1989 otherwise.

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