Hilda Assiyatun

Let G and H be simple graphs, the notation F → (G, H) means that for any red-blue coloring of all the edges of graph F implies F contains either a red copy of G

Let F, G and H be simple graphs. The notation F → (G, H) means that for any red-blue coloring on the edges of graph F, there exists either a red copy of G or a blue copy of H. A graph F is called a Ramsey (G, H)-minimal graph if it satisfies two conditions: (i) F → (G, H) and (ii) F − e ↛ (G, H) for any edge e of F. In this paper, we give some finite and infinite classes of Ramsey (C₄, K_1, n)-minimal graphs for any n ≥ 3.

Let F, G, and H be simple graphs. The notation F -> (G, H) means that if all the edges of F are arbitrarily colored by red or blue, then there always exists either a red subgraph G or a blue subgraph H. The size Ramsey number of graph G and H, denoted by (r) over cap (G, H) is the smallest integer k such that there is a graph F with k edges satisfying F -> (G, H). In this research, we will study a modified size Ramsey number, namely the connected size Ramsey number. In this case, we only consider connected graphs F satisfying the above properties. This connected size Ramsey number of G and H is denoted by (r) over cap (c)(G, H). We will derive an upper bound of (r) over cap (c)(nK(2), H), n >= 2 where H is 2P(m) or 2K(1,t), and find the exact values of (r) over cap (c)(nK(2), H), for some fixed n.

Finding all Ramsey (G, H)-minimal graphs for a certain pair of graphs G and H is an interesting and difficult problem. Even though, it is just for small graphs G and H. In this paper, we determine some Ramsey (P 3, P 6)-minimal graphs of small order. We also characterize all such Ramsey minimal graphs of order 6 by using their degree sequences. We prove that Ramsey (P 3, P 6)-minimal graphs have diameter at least two. We construct an infinite class of trees [6] which provides Ramsey (P 3, P 6)-minimal graphs.

Let F, G, and H be finite, simple, and undirected graphs. The notation F → (G, H) means that if the edge set of F is arbitrarily colored by red or blue, then there always exists either a red copy of G or a blue copy of H. The connected size Ramsey number r̂ c (G, H) is min{|E(F)| : F → (G, H), F is connected}. In this paper, we determine the connected size Ramsey numbers r̂ c (nK 2, mK 2), for n, m ≥ 2. Furthermore, an upper bound of r̂ c (nK 2, K 1,m ), for n ≥ 2 and m ≥ 3, and the exact value of r̂ c(nK 2, K 1,m ), for n = 2, 3, are presented.