A proof of a conjecture on matching-path connected size Ramsey number

For two graphs $G_1$ and $G_2$, the connected size Ramsey number ${\hat{r}}_c(G_1,G_2)$ is the smallest number of edges of a connected graph $G$ such that if each edge of $G$ is colored red or blue, then $G$ contains either a red copy of $G_1$ or a blue copy of $G_2$. Let $n{K}_2$ be a matching with $n$ edges and $P_4$ a path with four vertices. Rahadjeng, Baskoro, and Assiyatun [Procedia Comput. Sci. 74 (2015), 32-37] conjectured that ${\hat{r}}_c(n{K}_2,P_4)=3n-1$ if $n$ is even, and ${\hat{r}}_c(n{K}_2,P_4)=3n$ otherwise. We verify the conjecture in this short paper.