2 research outputs found
The connected size Ramsey number for matchings versus small disconnected graphs
Let F,βG,β and H be simple graphs. The notation Fβββ(G,βH) 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,β denoted by rΜ(G,βH) is the smallest integer k such that there is a graph F with k edges satisfying Fβββ(G,βH). 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 rΜc(G,βH). We will derive an upper bound of rΜc(nK2,βH),βnββ₯β2 where H is 2Pm or 2K1,βt,β and find the exact values of rΜc(nK2,βH),β for some fixed n.</p