Geodesic
On the Restricted Size Ramsey Number Involving a Path P3
Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 3, pp. 757-769

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For any pair of graphs G and H, both the size Ramsey number r̂(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,r̂ (G,H) ≤ r^∗ (G,H). When introducing the size Ramsey number for graph, Erdős et al. (1978) asked two questions; (1) Do there exist graphs G and H such that r̂ (G,H) attains the upper bound? and (2) Do there exist graphs G and H such that r̂ (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.
Keywords: restricted size Ramsey number, path, connected graph, star 
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Silaban, Denny Riama; Baskoro, Edy Tri; Uttunggadewa, Saladin. On the Restricted Size Ramsey Number Involving a Path P3. Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 3, pp. 757-769. http://geodesic.mathdoc.fr/item/DMGT_2019_39_3_a10/