Connected Size Ramsey Numbers for Matchings versus Cycles or Paths

Let F, G, and H be finite, simple and undirected graphs. The connected size Ramsey number r̂c(G, H) of graph G and H is the least integer k such that there is a connected graph F with k edges and 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. In this paper, we determine the connected size Ramsey number r̂c(2K2, Cn), for n ≥ 4, an upper bound of r̂c(nK2, P4), for n ≥ 2, and the exact value of r̂c(nK2, P4), for 2 ≤ n ≤ 5