scholarly journals Anti-Ramsey Numbers for Graphs with Independent Cycles

10.37236/174 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Zemin Jin ◽  
Xueliang Li
Keyword(s):  
Edge Coloring ◽  
Ramsey Number ◽  
Colored Graph ◽  
Ramsey Numbers ◽  
The Family

An edge-colored graph is called rainbow if all the colors on its edges are distinct. Let ${\cal G}$ be a family of graphs. The anti-Ramsey number $AR(n,{\cal G})$ for ${\cal G}$, introduced by Erdős et al., is the maximum number of colors in an edge coloring of $K_n$ that has no rainbow copy of any graph in ${\cal G}$. In this paper, we determine the anti-Ramsey number $AR(n,\Omega_2)$, where $\Omega_2$ denotes the family of graphs that contain two independent cycles.

scholarly journals Anti-Ramsey Numbers in Complete k-Partite Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Jili Ding ◽  
Hong Bian ◽  
Haizheng Yu
Keyword(s):  
Spanning Trees ◽  
Edge Coloring ◽  
Perfect Matchings ◽  
Ramsey Number ◽  
Hamilton Cycles ◽  
Ramsey Numbers ◽  
The Family

The anti-Ramsey number ARG,H is the maximum number of colors in an edge-coloring of G such that G contains no rainbow subgraphs isomorphic to H. In this paper, we discuss the anti-Ramsey numbers ARKp1,p2,…,pk,Tn, ARKp1,p2,…,pk,ℳ, and ARKp1,p2,…,pk,C of Kp1,p2,…,pk, where Tn,ℳ, and C denote the family of all spanning trees, the family of all perfect matchings, and the family of all Hamilton cycles in Kp1,p2,…,pk, respectively.

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 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 $2$.

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 Ramsey Numbers of Berge-Hypergraphs and Related Structures

10.37236/8892 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Nika Salia ◽  
Casey Tompkins ◽  
Zhiyu Wang ◽  
Oscar Zamora
Keyword(s):  
Edge Coloring ◽  
Ramsey Number ◽  
Uniform Hypergraph ◽  
Ramsey Numbers

For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists an injection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. Let the Ramsey number $R^r(BG,BG)$ be the smallest integer $n$ such that for any $2$-edge-coloring of a complete $r$-uniform hypergraph on $n$ vertices, there is a monochromatic Berge-$G$ subhypergraph. In this paper, we show that the 2-color Ramsey number of Berge cliques is linear. In particular, we show that $R^3(BK_s, BK_t) = s+t-3$ for $s,t \geq 4$ and $\max(s,t) \geq 5$ where $BK_n$ is a Berge-$K_n$ hypergraph. For higher uniformity, we show that $R^4(BK_t, BK_t) = t+1$ for $t\geq 6$ and $R^k(BK_t, BK_t)=t$ for $k \geq 5$ and $t$ sufficiently large. We also investigate the Ramsey number of trace hypergraphs, suspension hypergraphs and expansion hypergraphs.

Anti-Ramsey numbers for matchings in regular bipartite graphs

2017 ◽  
Vol 09 (02) ◽  
pp. 1750019 ◽  
Author(s):  
Zemin Jin ◽  
Oothan Nweit ◽  
Kaijun Wang ◽  
Yuling Wang
Keyword(s):  
Edge Coloring ◽  
Bipartite Graphs ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Number Formula

Let [Formula: see text] be a family of graphs. The anti-Ramsey number [Formula: see text] for [Formula: see text] in the graph [Formula: see text] is the maximum number of colors in an edge coloring of [Formula: see text] that does not have any rainbow copy of any graph in [Formula: see text]. In this paper, we consider the anti-Ramsey number for matchings in regular bipartite graphs and determine its value under several conditions.

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 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.

Mixed Ramsey Numbers Revisited

2003 ◽  
Vol 12 (5-6) ◽  
pp. 653-660
Author(s):  
C. C. Rousseau ◽  
S. E. Speed
Keyword(s):  
Asymptotic Behaviour ◽  
Ramsey Number ◽  
Open Problems ◽  
Free Graph ◽  
Ramsey Numbers ◽  
Dense Graphs

Given a graph Hwith no isolates, the (generalized) mixed Ramsey number is the smallest integer r such that every H-free graph of order r contains an m-element irredundant set. We consider some questions concerning the asymptotic behaviour of this number (i) with H fixed and , (ii) with m fixed and a sequence of dense graphs, in particular for the sequence . Open problems are mentioned throughout the paper.

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