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.

scholarly journals Degree Ramsey Numbers of Graphs

2012 ◽  
Vol 21 (1-2) ◽  
pp. 229-253 ◽  
Author(s):  
WILLIAM B. KINNERSLEY ◽  
KEVIN G. MILANS ◽  
DOUGLAS B. WEST
Keyword(s):  
Maximum Degree ◽  
Double Star ◽  
Ramsey Number ◽  
Bounded Degree ◽  
Ramsey Numbers ◽  
Colour Class ◽  
Bounded Degree Graphs ◽  
Edge Colourings ◽  
Edge Colouring ◽  
Monochromatic Copy

Let HG mean that every s-colouring of E(H) produces a monochromatic copy of G in some colour class. Let the s-colour degree Ramsey number of a graph G, written RΔ(G; s), be min{Δ(H): HG}. If T is a tree in which one vertex has degree at most k and all others have degree at most ⌈k/2⌉, then RΔ(T; s) = s(k − 1) + ϵ, where ϵ = 1 when k is odd and ϵ = 0 when k is even. For general trees, RΔ(T; s) ≤ 2s(Δ(T) − 1).To study sharpness of the upper bound, consider the double-starSa,b, the tree whose two non-leaf vertices have degrees a and b. If a ≤ b, then RΔ(Sa,b; 2) is 2b − 2 when a < b and b is even; it is 2b − 1 otherwise. If s is fixed and at least 3, then RΔ(Sb,b;s) = f(s)(b − 1) − o(b), where f(s) = 2s − 3.5 − O(s−1).We prove several results about edge-colourings of bounded-degree graphs that are related to degree Ramsey numbers of paths. Finally, for cycles we show that RΔ(C2k + 1; s) ≥ 2s + 1, that RΔ(C2k; s) ≥ 2s, and that RΔ(C4;2) = 5. For the latter we prove the stronger statement that every graph with maximum degree at most 4 has a 2-edge-colouring such that the subgraph in each colour class has girth at least 5.

scholarly journals Multicolour Bipartite Ramsey Number of Paths

10.37236/8458 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Matija Bucic ◽  
Shoham Letzter ◽  
Benny Sudakov
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Upper Bounds ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Over 40 Years ◽  
Monochromatic Copy

The k$-colour bipartite Ramsey number of a bipartite graph $H$ is the least integer $N$ for which every $k$-edge-coloured complete bipartite graph $K_{N,N}$ contains a monochromatic copy of $H$. The study of bipartite Ramsey numbers was initiated over 40 years ago by Faudree and Schelp and, independently, by Gyárfás and Lehel, who determined the $2$-colour bipartite Ramsey number of paths. Recently the $3$-colour Ramsey number of paths and (even) cycles, was essentially determined as well. Improving the results of DeBiasio, Gyárfás, Krueger, Ruszinkó, and Sárközy, in this paper we determine asymptotically the $4$-colour bipartite Ramsey number of paths and cycles. We also provide new upper bounds on the $k$-colour bipartite Ramsey numbers of paths and cycles which are close to being tight.

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 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 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 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.

Star Versus Two Stripes Ramsey Numbers and a Conjecture of Schelp

2012 ◽  
Vol 21 (1-2) ◽  
pp. 179-186 ◽  
Author(s):  
ANDRÁS GYÁRFÁS ◽  
GÁBOR N. SÁRKÖZY
Keyword(s):  
Minimum Degree ◽  
Ramsey Number ◽  
Regularity Lemma ◽  
Ramsey Numbers ◽  
Standard Application ◽  
Star Versus ◽  
Ramsey Type ◽  
Weakened Form

R. H. Schelp conjectured that if G is a graph with |V(G)| = R(Pn, Pn) such that δ(G) > $$\frac{3|V(G)|}{ 4}$, then in every 2-colouring of the edges of G there is a monochromatic Pn. In other words, the Ramsey number of a path does not change if the graph to be coloured is not complete but has large minimum degree.Here we prove Ramsey-type results that imply the conjecture in a weakened form, first replacing the path by a matching, showing that the star-matching–matching Ramsey number satisfying R(Sn, nK2, nK2) = 3n − 1. This extends R(nK2, nK2) = 3n − 1, an old result of Cockayne and Lorimer. Then we extend this further from matchings to connected matchings, and outline how this implies Schelp's conjecture in an asymptotic sense through a standard application of the Regularity Lemma.It is sad that we are unable to hear Dick Schelp's reaction to our work generated by his conjecture.

The Induced Size-Ramsey Number of Cycles

1995 ◽  
Vol 4 (3) ◽  
pp. 217-239 ◽  
Author(s):  
P. E. Haxell ◽  
Y. Kohayakawa ◽  
T. Łuczak
Keyword(s):  
Random Graphs ◽  
Ramsey Number ◽  
Regularity Lemma ◽  
Induced Subgraph ◽  
Natural Notion ◽  
Number Of Cycles

For a graph H and an integer r ≥ 2, the induced r-size-Ramsey number of H is defined to be the smallest integer m for which there exists a graph G with m edges with the following property: however one colours the edges of G with r colours, there always exists a monochromatic induced subgraph H′ of G that is isomorphic to H. This is a concept closely related to the classical r-size-Ramsey number of Erdős, Faudree, Rousseau and Schelp, and to the r-induced Ramsey number, a natural notion that appears in problems and conjectures due to, among others, Graham and Rödl, and Trotter. Here, we prove a result that implies that the induced r-size-Ramsey number of the cycle Cℓ is at most crℓ for some constant cr that depends only upon r. Thus we settle a conjecture of Graham and Rödl, which states that the above holds for the path Pℓ of order ℓ and also generalise in part a result of Bollobás, Burr and Reimer that implies that the r-size Ramsey number of the cycle Cℓ is linear in ℓ Our method of proof is heavily based on techniques from the theory of random graphs and on a variant of the powerful regularity lemma of Szemerédi.

scholarly journals Degree Ramsey Numbers of Closed Blowups of Trees

10.37236/2526 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Paul Horn ◽  
Kevin G. Milans ◽  
Vojtěch Rödl
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Edge Coloring ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Delta G ◽  
Monochromatic Copy

The degree Ramsey number of a graph $G$, denoted $R_\Delta(G;s)$, is $\min\{\Delta(H)\colon\, H\stackrel{s}{\to} G\}$, where $H\stackrel{s}{\to} G$ means that every $s$-edge-coloring of $H$ contains a monochromatic copy of $G$.  The closed $k$-blowup of a graph is obtained by replacing every vertex with a clique of size $k$ and every edge with a complete bipartite graph where both partite sets have size $k$.  We prove that there is a function $f$ such that $R_\Delta(G;s) \le f(\Delta(G), s)$  when $G$ is a closed blowup of a tree.

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 $R(P_{3}, P_{5}, C_{k})=k+1$.

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