Restricted Size Ramsey Number Research Articles

Restricted size Ramsey numbers and restricted size Ramsey minimal graphs for matching versus wheel upto order six

The notion of the restricted size Ramsey number arises from the amalgamation of the theory of Ramsey numbers and the size Ramsey number for graphs. The concept of the (restricted) size Ramsey number of small graphs was first introduced by Harary and Miller in 1983. They derived specific values for certain pairs of small graphs, each having a maximum count of four vertices. In that same year, R. J. Faudree and J. Sheehan furthered the research and expanded the findings to encompass all the pairs of small graphs with a maximum order of four. The restricted size Ramsey number and size Ramsey number for all pairs of small forests of maximum order five was given by Lortz and Mengersen in 1998. In this study, our research endeavours to advance the understanding of the restricted size Ramsey number by ascertaining the restricted size Ramsey numbers and restricted size Ramsey minimal graphs involving 2K2 and wheel graph upto order six. More specifically, we find r * (2K2, W3) = r* (2K2, W4) = 15 and r * (2K2, W5) = 19.

Restricted Size Ramsey Number for Matching versus Tree and Triangle Unicyclic Graphs of Order Six

Let F, G, and H be simple graphs. The graph F arrows (G,H) if for any red-blue coloring on the edge of F, we find either a red-colored graph G or a blue-colored graph H in F. The Ramsey number r(G,H) is the smallest positive integer r such that a complete graph Kr arrows (G,H). The restricted size Ramsey number r∗(G,H) is the smallest positive integer r∗ such that there is a graph F, of order r(G,H) and with the size r∗, satisfying F arrows (G,H). In this paper we give the restricted size Ramsey number for a matching of two edges versus tree and triangle unicyclic graphs of order six.

Restricted size Ramsey number for 2K 2 versus disconnected graphs of order six

Given simple graphs F, G, and H. We say F arrows (G, H) if for any red-blue coloring of the edge of F, we find either a red-colored graph G or a blue-colored graph H. The Ramsey number r(G, H) is the smallest positive integer r such that a complete graph K r arrows (G, H). The size Ramsey number is the smallest positive integer such that a graph F with the size of arrows (G, H). The restricted size Ramsey number is the smallest positive integer r ∗ such that a graph F, of order r(G, H) with the size of r ∗, arrows (G, H). In this paper we give the restricted size Ramsey number of a matching of two edges and any disconnected graphs of order six with no isolates.

On the restricted size Ramsey number for P3 versus dense connected graphs

Let F , G and H be simple graphs. A graph F is said a ( G,H )-arrowing graph if in any red-blue coloring of edges of F we can find a red G or a blue H . The size Ramsey number of G and H , ŕ ( G , H ), is the minimum size of F . If the order of F equals to the Ramsey number of G and H , r ( G,H ), then the minimum size of F is called the restricted size Ramsey number of G and H , r * ( G,H ). The Ramsey number of G and H , r ( G,H ), is the minimum order of F . In this paper, we study the restricted size number involving a P 3 . The value of r * ( P 3 , K n ) has been given by Faudree and Sheehan. Here, we examine r * ( P 3 , H ) where H is dense connected graph.

Restricted size Ramsey number for P3 versus cycle

Let F , G and H be simple graphs. We say F → ( G , H ) if for every 2-coloring of the edges of F there exists a red copy of G or a blue copy of H in F . The Ramsey number r ( G,H ) is defined as r ( G,H ) = min{| V ( F )|: F → ( G,H )}, while the restricted size Ramsey number r * ( G,H ) is defined as r * ( G,H ) = min{| E ( F )|: F → ( G,H ),| V ( F )| = r ( G,H )}. In this paper we determine previously unknown restricted size Ramsey numbers r * ( P 3 , C n ) for 7 ≤ n ≤ 12. We also give new upper bound r * ( P 3 , C n ) ≤ 2 n -2 for n ≥ 10 and n is even.

Restricted Size Ramsey Number Involving Matching and Graph of Order Five

Harary and Miller (1983) started the research on the (restricted) size Ramsey number for a pair of small graphs. They obtained the values for some pairs of small graphs with order not more than four. In the same year, Faudree and Sheehan continued the research and extended the result to all pairs of small graphs with order not more than four. Moreover, in 1998, Lortz and Mengenser gave the size Ramsey number and the restricted size Ramsey number for all pairs of small forests with order not more than five. Recently, we gave the restricted size Ramsey number for a path of order three and any connected graph of order five. In this paper, we continue the research on the (restricted) size Ramsey number involving small graphs by investigating the restricted size Ramsey number for matching with two edges versus any graph of order five with no isolates.

On the restricted size Ramsey number involving a path P_3

On the restricted size Ramsey number involving a path P_3

Restricted Size Ramsey Number for 2K2 versus Dense Connected Graphs of Order Six

Let G and H be simple graphs. The Ramsey number r(G, H) for a pair of graphs G and H is the smallest number r such that any red-blue coloring of the edges of Kr contains a red subgraph G or a blue subgraph H. The size Ramsey number for a pair of graphs G and H is the smallest number such that there exists a graph F with size satisfying the property that any red-blue coloring of the edges of F contains a red subgraph G or a blue subgraph H. Additionally, if the order of F in the size Ramsey number equals r(G, H), then it is called the restricted size Ramsey number. In 1972, Chvátal and Harary gave the Ramsey number for 2K2 versus any graph H with no isolates. In 1983, Harary and Miller started the investigation of the (restricted) size Ramsey number for some pairs of small graphs with order at most four. In 1983, Faudree and Sheehan continued the investigation and summarized the complete results on the (restricted) size Ramsey number for all pairs of small graphs with order at most four. In 1998, Lortz and Mengenser gave both the size Ramsey number and the restricted size Ramsey number for all pairs of small forests with order at most five. Lately, we investigate the restricted size Ramsey number for 2K2 versus all connected graphs of order five. In this work, we continue the investigation on the restricted size Ramsey number for a pair of small graphs. In particularly, for 2K2 versus dense connected graph of order six.

Restricted size Ramsey number for path of order three versus graph of order five

Let $G$ and $H$ be simple graphs. The Ramsey number for a pair of graph $G$ and $H$ is the smallest number $r$ such that any red-blue coloring of edges of $K_r$ contains a red subgraph $G$ or a blue subgraph $H$. The size Ramsey number for a pair of graph $G$ and $H$ is the smallest number $\hat{r}$ such that there exists a graph $F$ with size $\hat{r}$ satisfying the property that any red-blue coloring of edges of $F$ contains a red subgraph $G$ or a blue subgraph $H$. Additionally, if the order of $F$ in the size Ramsey number is $r(G,H)$, then it is called the restricted size Ramsey number. In 1983, Harary and Miller started to find the (restricted) size Ramsey number for any pair of small graphs with order at most four. Faudree and Sheehan (1983) continued Harary and Miller's works and summarized the complete results on the (restricted) size Ramsey number for any pair of small graphs with order at most four. In 1998, Lortz and Mengenser gave both the size Ramsey number and the restricted size Ramsey number for any pair of small forests with order at most five. To continue their works, we investigate the restricted size Ramsey number for a path of order three versus connected graph of order five.

On The Restricted Size Ramsey Number

Abstract Let F , G , and H be simple graphs. We say F → ( G , H ) if for every 2-coloring of the edges of F there exists a monochromatic G or H in F . The Ramsey number r ( G , H ) is defined as min {| V ( F )|: F → ( G , H )}, while the restricted size Ramsey number r ∗( G , H ) is defined as min {| E ( F )|: F → ( G , H ), | V ( F ) | = r ( G , H )}. In this paper, we give lower and upper bounds for the restricted size Ramsey number for a path P 3 versus cycles Cn .

Size Ramsey numbers for small‐order graphs

AbstractA table of the size Ramsey number or the restricted size Ramsey number for all pairs of graphs with at most four vertices and no isolated vertices is given.