Restricted size Ramsey number for $P_3$ versus cycles

Let $F$, $G$ and $H$ be simple graphs. We say $F\to (G,H)$ if for every $2$-coloring of the edges of $F$ there exists a monochromatic $G$ or $H$ in $F$. The Ramsey number $r(G,H)$ is defined as $r(G, H) = min{|V (F)|: F \rightarrow (G, H)}$,whiletherestrictedsizeRamseynumber$r^{}(G, H)$ is defined as $r^{}(G, H) = min{|E (F)|: F \rightarrow (G, H) , |V (F) | = r(G, H)}$. In this paper we determine previously unknown restricted size Ramsey numbers $r^{\ast }(P_3,C_n)$ for $7\le n\le 12$. We also give new upper bound $r^{\ast }(P_3,C_n)\le 2n-2$ for even $n\ge 8$.