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

Given two graphs $G_1, G_2$, the connected size Ramsey number ${\hat{r}}_c(G_1,G_2)$ 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 $G_1$ or a blue copy of $G_2$. Concentrating on ${\hat{r}}_c(nK_2,G_2)$ where $nK_2$ is a matching, we generalise and improve two previous results as follows. Vito, Nabila, Safitri, and Silaban obtained the exact values of ${\hat{r}}_c(nK_2,P_3)$ for $n=2,3,4$. We determine its exact values for all positive integers $n$. Rahadjeng, Baskoro, and Assiyatun proved that ${\hat{r}}_c(nK_2,C_4)\le 5n-1$ for $n\ge 4$. We improve the upper bound from $5n-1$ to $\lfloor (9n-1)/2 \rfloor$. In addition, we show a result which has the same flavour and has exact values: ${\hat{r}}_c(nK_2,C_3)=4n-1$ for all positive integers $n$.

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