On the multicolor Ramsey number of a graph with m edges

Kathleen Johst; Yury Person

doi:10.1016/j.disc.2016.05.021

On the Multicolor Ramsey Number for 3-Paths of Length Three

The Electronic Journal of Combinatorics ◽

10.37236/6670 ◽

2017 ◽

Vol 24 (1) ◽

Cited By ~ 2

Author(s):

Tomasz Łuczak ◽

Joanna Polcyn

Keyword(s):

Ramsey Number ◽

Uniform Hypergraph ◽

Multicolor Ramsey Number

We show that if we color the hyperedges of the complete $3$-uniform hypergraph on $2n+\sqrt{18n+1}+2$ vertices with $n$ colors, then one of the color classes contains a loose path of length three.

Download Full-text

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

The Electronic Journal of Combinatorics ◽

10.37236/1081 ◽

2006 ◽

Vol 13 (1) ◽

Cited By ~ 6

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

Download Full-text

New upper bound for multicolor Ramsey number of odd cycles

Discrete Mathematics ◽

10.1016/j.disc.2018.09.014 ◽

2019 ◽

Vol 342 (1) ◽

pp. 217-220 ◽

Cited By ~ 1

Author(s):

Qizhong Lin ◽

Weiji Chen

Keyword(s):

Upper Bound ◽

Ramsey Number ◽

Multicolor Ramsey Number ◽

Odd Cycles

Download Full-text

On Multicolor Ramsey Number of Paths Versus Cycles

The Electronic Journal of Combinatorics ◽

10.37236/511 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 2

Author(s):

Gholam Reza Omidi ◽

Ghaffar Raeisi

Keyword(s):

Lower Bound ◽

Positive Integer ◽

Complete Graph ◽

Ramsey Number ◽

Subgraph Isomorphic ◽

Multicolor Ramsey Number

Let $G_1, G_2, G_3, \ldots , G_t$ be graphs. The multicolor Ramsey number $R(G_1, G_2, \ldots, G_t)$ is the smallest positive integer $n$ such that if the edges of a complete graph $K_n$ are partitioned into $t$ disjoint color classes giving $t$ graphs $H_1,H_2,\ldots,H_t$, then at least one $H_i$ has a subgraph isomorphic to $G_i$. In this paper, we provide the exact value of $R(P_{n_1}, P_{n_2},\ldots, P_{n_t},C_k)$ for certain values of $n_i$ and $k$. In addition, the exact values of $R(P_5,C_4,P_k)$, $R(P_4,C_4,P_k)$, $R(P_5,P_5,P_k)$ and $R(P_5,P_6,P_k)$ are given. Finally, we give a lower bound for $R(P_{2n_1}, P_{2n_2},\ldots, P_{2n_t})$ and we conjecture that this lower bound is the exact value of this number. Moreover, some evidence is given for this conjecture.

Download Full-text

Ramsey Number of Wheels Versus Cycles and Trees

Canadian Mathematical Bulletin ◽

10.4153/cmb-2015-057-5 ◽

2016 ◽

Vol 59 (01) ◽

pp. 190-196 ◽

Cited By ~ 1

Author(s):

Ghaffar Raeisi ◽

Ali Zaghian

Keyword(s):

Positive Integer ◽

Complete Graph ◽

Ramsey Number ◽

Subgraph Isomorphic ◽

Multicolor Ramsey Number ◽

Arbitrary Graphs

Abstract Let G 1, G 2 , …, Gt be arbitrary graphs. The Ramsey number R(G 1 , G 2, …, Gt ) is the smallest positive integer n such that if the edges of the complete graph Kn are partitioned into t disjoint color classes giving t graphs H 1, H 2, …, H t, then at least one Hi has a subgraph isomorphic to Gi. In this paper, we provide the exact value of the R(Tn, Wm) for odd m, n ≥ m−1, where T n is either a caterpillar, a tree with diameter at most four, or a tree with a vertex adjacent to at least leaves, and W n is the wheel on n + 1 vertices. Also, we determine R(C n, W m) for even integers n and m, n ≥ m + 500, which improves a result of Shi and confirms a conjecture of Surahmat et al. In addition, the multicolor Ramsey number of trees versus an odd wheel is discussed in this paper.

Download Full-text

New Lower Bound for Multicolor Ramsey Numbers for Even Cycles

The Electronic Journal of Combinatorics ◽

10.37236/1980 ◽

2005 ◽

Vol 12 (1) ◽

Cited By ~ 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.

Download Full-text

A Note on On-Line Ramsey Numbers for Some Paths

Mathematics ◽

10.3390/math9070735 ◽

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.

Download Full-text