Let F, G, and H be finite, simple, and undirected graphs. The notation F → (G, H) means that if the edge set of F is arbitrarily colored by red or blue, then there always exists either a red copy of G or a blue copy of H. The connected size Ramsey number r̂c(G, H) is min{|E(F)| : F → (G, H), F is connected}. In this paper, we determine the connected size Ramsey numbers r̂c(nK2, mK2), for n, m ≥ 2. Furthermore, an upper bound of r̂c(nK2, K1,m), for n ≥ 2 and m ≥ 3, and the exact value of r̂c(nK2, K1,m), for n = 2, 3, are presented.
Topics
Ramsey's theory
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