For any pair of graphs G and H, both the size Ramsey number r(G, H) and the restricted size Ramsey number r ∗ (G, H) are bounded above by the size of the complete graph with order equals to the Ramsey number r(G, H), and bounded below by e(G) + e(H) − 1. Moreover, trivially, r(G, H) ≤ r ∗ (G, H). When introducing the size Ramsey number for graph, Erdös et al. (1978) asked two questions; (1) Do there exist graphs G and H such that r(G, H) attains the upper bound? and (2) Do there exist graphs G and H such that r(G, H) is significantly less than the upper bound? In this paper we consider the restricted size Ramsey number r ∗ (G, H). We answer both questions above for r ∗ (G, H) when G = P 3 and H is a connected graph.
- Connected graph
- Restricted size ramsey number