Related papers: Ramsey numbers for trees II

Given a graph $G$ and a positive integer $k$, the \emph{Gallai-Ramsey number} is defined to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) copy of $G$ or a…

For two graphs $G,H$, the \emph{Ramsey number} $r(G,H)$ is the minimum integer $n$ such that any red/blue edge-coloring of $K_n$ contains either a red copy of $G$ or a blue copy of $H$. For two graphs $G,H$, the \emph{Gallai-Ramsey number}…

Chv\'atal showed that for any tree $T$ with $k$ edges the Ramsey number $R(T,n)=k(n-1)+1$ ("Tree-complete graph Ramsey numbers." Journal of Graph Theory 1.1 (1977): 93-93). For $r=3$ or $4$, we show that, if $T$ is an $r$-uniform…

Chen et al. (2004) strongly conjectured that R(Tn,Wm)=2n-1 if the maximum degree of Tn is small and m is even. Related to the conjecture, it is interesting to know for which tree Tn, we have R(Tn,Wm) > 2n-1 for even m. In this paper, we…

Let $G$ be a complete multi-partite graph of order $n$. In this paper, we consider the anti-Ramsey number $ar(G,\mathcal{T}_{q})$ with respect to $G$ and the set $\mathcal{T}_{q}$ of trees with $q$ edges, where $2\le q\le n-1$. For the case…

Let $G_1$ and $G_2$ be two given graphs. The Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\overline{G}$ contains a $G_2$. We denote by $P_n$ the path on…

Let $r(G,H)$ be the smallest integer $N$ such that for any $2$-coloring (say, red and blue) of the edges of $K\_n$, $n\geqslant N$, there is either a red copy of $G$ or a blue copy of $H$. Let $K\_n-K\_{1,s}$ be the complete graph on $n$…

We call a subgraph of an edge-colored graph rainbow subgraph, if all of its edges have different colors. The anti-Ramsey number of a graph $G$ in a complete graph $K_{n}$, denoted by $ar(K_{n}, G)$, is the maximum number of colors in an…

In this paper we study Ramsey numbers for trees of diameter 3 (bistars) vs., respectively, trees of diameter 2 (stars), complete graphs, and many complete graphs. In the case of bistars vs. many complete graphs, we determine this number…

Given graphs $ F_1, F_2$ and $G$, we say that $G$ is Ramsey for $(F_1,F_2)$ and we write $G\rightarrow(F_1, F_2)$, if for every edge coloring of $G$ by red and blue, there is either a red copy of $F_1$ or a blue copy of $F_2$ in $G$. The…

For given graphs $G_{1}, G_{2}$ and $G$, let $G\rightarrow (G_{1}, G_{2})$ denote that each red-blue-coloring of $E(G)$ yields a red copy of $G_{1}$ or a blue copy of $G_{2}$. Arag{\~a}o, Marciano and Mendon{\c c}a [L. Arag{\~a}o, J. Pedro…

For every $k\ge 2$ and $\Delta$, we prove that there exists a constant $C_{\Delta,k}$ such that the following holds. For every graph $H$ with $\chi(H)=k$ and every tree with at least $C_{\Delta,k}|H|$ vertices and maximum degree at most…

The generalized Ramsey number $R(H, K)$ is the smallest positive integer $n$ such that for any graph $G$ with $n$ vertices either $G$ contains $H$ as a subgraph or its complement $\overline{G}$ contains $K$ as a subgraph. Let $T_n$ be a…

A double star $S(n,m)$ is the graph obtained by joining the center of a star with $n$ leaves to a center of a star with $m$ leaves by an edge. Let $r(S(n,m))$ denote the Ramsey number of the double star $S(n,m)$. In 1979 Grossman, Harary…

In this paper, we obtain upper bounds for the geometric Ramsey numbers of trees. We prove that $R_c(T_n,H_m)=(n-1)(m-1)+1$ if $T_n$ is a caterpillar and $H_m$ is a Hamiltonian outerplanar graph on $m$ vertices. Moreover, if $T_n$ has at…

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a…

Let $n,r,k,s$ be positive integers with $n,k\ge 2$. The generalized Ramsey number $R(n,r;k,s)$ is the smallest positive integer $p$ such that for every graph $G$ of order $p$, either $G$ contains a subgraph induced by $n$ vertices with at…

Given bipartite graphs $G_1, \ldots, G_n$, the bipartite Ramsey number $BR(G_1,\ldots, G_n)$ is the last integer $b$ such that any complete bipartite graph $K_{b,b}$ with edges coloured with colours $1,2,\ldots,n$ contains a copy of some…

The balanced double star on $2n+2$ vertices, denoted $S_{n,n}$, is the tree obtained by joining the centers of two disjoint stars each having $n$ leaves. Let $R_r(G)$ be the smallest integer $N$ such that in every $r$-coloring of the edges…

We obtain some new upper bounds on the Ramsey numbers of the form $R(\underbrace{C_4,\ldots,C_4}_m,G_1,\ldots,G_n)$, where $m\ge 1$ and $G_1,\ldots,G_n$ are arbitrary graphs. We focus on the cases of $G_i$'s being complete, star $K_{1,k}$…