Given a graph F, let L ( F ) be a fixed finite family of graphs consisting of a C 4 and some bipartite graphs relying on an s-partite subgraph partitioning of edges of F. Define ( m , n , a , b )-graph by an m × n bipartite graph with n ≥ m such ...

For graphs G and H, the Ramsey number r k , p ( G ; H ) is defined as the smallest integer N such that any edge-coloring of K N by k + p colors contains either a monochromatic G in one of the first k colors, or a monochromatic H in one of the ...

For a graph G, let $R(\mathcal{C}(nG))$ denote the least N such that every 2-colouring of the edges of $K_N$ contains a monochromatic copy of nG in a monochromatic connected subgraph, where nG denotes n vertex disjoint copies of G. Gyárfás and Sárközy (J Graph ...$\mathcal{}{}_{\mathrm{}}\mathrm{}\mathrm{}$$\mathrm{}$$\mathcal{}{}_{}{}^{\mathrm{}}\mathrm{}\mathrm{}$$\mathrm{}$${}_{}$${}_{}$${}_{}$$\mathcal{}$

In this paper, we study the Ramsey properties for V-shaped posets. A V-shaped poset V m , n can be obtained by identifying the minimal elements of two chains on m + 1 and n + 1 elements, respectively. Let P 1 , P 2 , … , P k be posets. The ...

For a graph H and an integer k ≥ 1, let r ( H ; k ) and r ℓ ( H ; k ) denote the k-color Ramsey number and list Ramsey number of H, respectively. Alon, Bucić, Kalvari, Kuperwasser and Szabó in 2021 initiated the systematic study of list Ramsey ...

For a graph G, the Ramsey number R k ( G ) is defined as the minimum r such that any k-edge coloring of K r contains a monochromatic G. A Gallai k-coloring is a k-edge coloring of a complete graph without rainbow triangle, where a rainbow ...

Given two graphs G and H, the generalk-colored Gallai–Ramsey number${\text{gr}}_k(G:H)$ is defined to be the minimum integer m such that every k-coloring of the complete graph on m vertices contains either a rainbow copy of G or a monochromatic copy of H. ...${}_{}$${}_{{}_{}}$

For graphs H 1 , H 2 by r ⁎ ( H 1 , H 2 ) we denote the minimum number of edges in a graph G on r ( H 1 , H 2 ) vertices such that G → ( H 1 , H 2 ). We show that for each pair of natural numbers k , n, k ≤ n, where k is odd and n is large enough,...

For an arbitrary graph G, a hypergraph H is called Berge-G if there is an injection i : V ( G ) ⟶ V ( H ) and a bijection ψ : E ( G ) ⟶ E ( H ) such that for each e = u v ∈ E ( G ), we have { i ( u ) , i ( v ) } ⊆ ψ ( e ). We denote by B r G, the ...

Let G 1 , G 2 , … , G c be arbitrary graphs. By G → ( G 1 , G 2 , … , G c ), we mean if the edges of G are partitioned into c disjoint color classes giving c graphs H 1 , H 2 , … , H c, then at least one H i has a subgraph isomorphic to G i. The ...

Let r(s, t) be the classical 2-color Ramsey number; that is, the smallest integer n such that any edge 2-colored $K_n$ contains either a monochromatic $K_s$ of color 1 or $K_t$ of color 2. Define the signed Ramsey number$r_{\pm }(s,t)$ to be the smallest integer n ...${}_{}$${}_{}$${}_{}$

• Let Γ be (di)graph. A family P of sub(di)graphs of Γ is called a path cover (resp. a path partition) of Γ if ∪ p ϵ P V ( P ) = V ( Γ ) (resp. ∪ p ϵ P V ( P ) = V ( Γ )) and every element of P is a path or a directed path. The minimum ...

Recently, the authors [Electron. J. Combin. 29 (2022), Paper No. 4.8] gave Ramsey-type results for the path cover/partition number of graphs. In this paper, we continue the research about them focusing on digraphs, and find a relationship between ...

We find the exact value of the Ramsey number R ( C 2 ℓ , K 1 , n ), when ℓ and n = O ( ℓ 10 / 9 ) are large. Our result is closely related to the behaviour of Turán number ex ( N , C 2 ℓ ) for an even cycle whose length grows quickly ...

Given two graphs G and H, the Ramsey number r ( G , H ) is the minimum integer N such that any red-blue coloring of the edges of K N contains either a red copy of G or a blue copy of H. Let v ( G ) denote the number of vertices of G, ...

For graphs F, G and H, let $F\to (G,H)$ signify that any red-blue edge coloring of F contains either a red G or a blue H, hence the Ramsey number R(G, H) is the smallest r such that $K_r\to (G,H)$. Define $K_t$ as the surplus clique of (G, H) if $K_r\backslash K_t\to (G,H)$, ...$$$\mathrm{}$${}_{}$$\frac{}{\mathrm{}}$

A graph $\mathcal{H}=(W,E_{\mathcal{H}})$ is said to have bandwidth at most b if there exists a labeling of W as $w_1,w_2,\cdots ,w_n$ such that $|i-j|\le b$ for every edge $w_i{w}_j\in E_{\mathcal{H}}$. We say that $\mathcal{H}$ is a balanced $(\beta ,\mathrm{\Delta })$-graph if it is a bipartite graph with bandwidth at most $\beta |W|$ and maximum ...$\mathrm{}$$\mathrm{}$${}^{\mathrm{}}\mathrm{}{}^{\mathrm{}}\mathrm{}{}^{\mathrm{}}\mathrm{}$$\mathrm{}$$\mathrm{}$$\mathrm{}$$\mathrm{}$$\mathcal{}$$$\begin{array}{c}\hfill \mathcal{}\mathcal{}{}_{}\mathrm{}\end{array}$$${}_{}$$\mathrm{}$${}_{}{}_{}{}_{}\mathrm{}\mathrm{}$$\mathrm{}$$\mathrm{}$${}_{\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}\mathrm{}}\mathrm{}\mathrm{}$

Given two graphs G and H, the Ramsey number R(G, H) is the minimum integer N such that any coloring of the edges of $K_N$ in red or blue yields a red G or a blue H. Let $\chi (G)$ be the chromatic number of G, and s(G) denote the chromatic surplus of G, ...$$$\mathrm{}\mathrm{}$${}_{}$$\mathrm{}$${}_{}$${}_{}$$\mathrm{}{}_{}$$\mathrm{}$$\mathrm{}$${}_{}$$\mathrm{}{}_{}$${}_{}$$\mathrm{}{}_{}$$\mathrm{}$$\mathrm{}$${}_{}$${}_{}{}_{}$${}_{}{}_{}$${}_{}$${}_{}$$\mathrm{}$$$${}_{}{}_{}$$\mathrm{}$

Given a positive integer r, the r-color size-Ramsey number of a graph H, denoted by R ˆ ( H , r ), is the smallest number of edges in a graph G such that in any edge coloring of G with r colors, G contains a monochromatic copy of H. ...

For two graphs G and H, let r(G, H) and $r_{\ast }(G,H)$ denote the Ramsey number and star-critical Ramsey number of G versus H, respectively. In 1996, Li and Rousseau proved that $r(K_m,F_{t,n})=tn(m-1)+1$ for $m\ge 3$ and sufficiently large n, where $F_{t,n}=K_1+n{K}_t$. ...${}_{\mathrm{}}{}_{\mathrm{}}\mathrm{}\mathrm{}$$\mathrm{}$${}_{}{}_{\mathrm{}}{}_{\mathrm{}}\mathrm{}\mathrm{}$$\mathrm{}$${}_{}{}_{}\mathrm{}$${}_{\mathrm{}}{}_{}\mathrm{}$$\mathrm{}\mathrm{}$$\mathrm{}$${}_{}{}_{\mathrm{}}{}_{\mathrm{}}\mathrm{}\mathrm{}$$\mathrm{}$${}_{\mathrm{}}{}_{}$

It is shown that the Ramsey number $r(C_{2m+1},C_{2m+1},K_t)$ is between $\mathrm{\Omega }(t^{1+\frac{2}{2m-1}}/{(logt)}^{\frac{4}{2m-1}})$ and $O(t^{(1+\frac{1}{m}{)}^2}/{(logt)}^{\frac{2m+1}{m^2}})$ for fixed m and large t.