Restricted size Ramsey number for path of order three versus graph of order five

Let G and H be simple graphs. The Ramsey number r(G,H) for a pair of graphs G and H is the smallest number r such that any red-blue coloring of the edges of Kr contains a red subgraph G or a blue subgraph H . The size Ramsey number r̂(G,H) for a pair of graphs G and H is the smallest number r̂ such that there exists a graph F with size r̂ satisfying the property that any red-blue coloring of the edges of F contains a red subgraph G or a blue subgraph H . Additionally, if the order of F in the size Ramsey number equals r(G,H), then it is called the restricted size Ramsey number. In 1983, Harary and Miller started to find the (restricted) size Ramsey numbers for pairs of small graphs with orders at most four. Faudree and Sheehan (1983) continued Harary and Miller’s works and summarized the complete results on the (restricted) size Ramsey numbers for pairs of small graphs with orders at most four. In 1998, Lortz and Mengenser gave both the size Ramsey numbers and the restricted size Ramsey numbers for pairs of small forests with orders at most five. To continue their works, we investigate the restricted size Ramsey numbers for a path of order three versus any connected graph of order five.