Connected size Ramsey numbers of matchings versus a small path or cycle

Given two graphs G1, G2, the connected size Ramsey number rc(G1, G2) is defined to be the minimum number of edges of a connected graph G, such that for any red-blue edge colouring of G, there is either a red copy of G1 or a blue copy of G2. Concentrating on rc(nK2, G2) where nK2 is a matching, we generalise and improve two previous results as follows. Vito, Nabila, Safitri, and Silaban (J. Phys. Conf. Ser., 2021) obtained the exact values of rc(nK2, P3) for n = 2, 3, 4. We determine its exact values for all positive integers n. Rahadjeng, Baskoro, and Assiyatun (Proc. Indian Acad. Sci.: Math. Sci., 2017) proved that rc(nK2, C4) <= 5n - 1 for n >= 4. We improve the upper bound from 5n - 1 to & LeftFloor;(9n - 1)/2 & RightFloor;. In addition, we show a result which has the same flavour and has exact values: rc(nK2, C3) = 4n - 1 for all positive integers n.