Restricted size Ramsey number for P3 versus dense connected graphs of order six

Abstract

Let F, G, and H be simple graphs. We say F → (G, H) if for every 2-coloring of the edges of F there exist a monochromatic G or H in F. The Ramsey number r(G, H) is min {v(F): F → (G, H)}, the size Ramsey number r*(G, H) is min {e(F): F → (G, H)}, and the restricted size Ramsey number r*(G, H) is min {e(F): F → (G, H), v(F) = r(G, H)}. In 1972, Chvá tal and Harary gave the Ramsey number for P3 versus any graph H with no isolates. In 1983, Harary and Miller started the investigation of the (restricted) size Ramsey number for some pairs of small graphs with order at most four. In 1983, Faudree and Sheehan continued the investigation and summarized the complete results on the (restricted) size Ramsey number for all pairs of small graphs with order at most four. In 1998, Lortz and Mengenser gave both the size Ramsey number and the restricted size Ramsey number for all pairs of small forests with order at most five. Lately, we investigate the restricted size Ramsey number for a path of order three versus all connected graphs of order five. In this work, we continue the investigation on the restricted size Ramsey number for a pair of small graphs. In particularly, for a path of order three versus connected graphs of order six.