  Let $F$, $G$ and $H$ be simple graphs. We say $F \rightarrow (G, H)$ 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 $r(G, H) = min\{|V (F)|: F
\rightarrow (G, H)\}$, while the restricted size Ramsey number $r^{*}(G, H)$ 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 $r^{*}(P_3, C_n)$ for $7 \leq n \leq 12$. We also give new upper bound
$r^{*}(P_3, C_n) \leq 2n-2$ for even $n \geq 8$.
