The connected size Ramsey number for matchings versus small disconnected graphs

Abstract

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 ( 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 c ( G , H ). We will derive an upper bound of r c ( n K 2 , H ), n ≥ 2 where H is 2 P m or 2 K 1, t , and find the exact values of r c ( n K 2 , H ), for some fixed n .