STAR-CRITICAL RAMSEY NUMBERS FOR CYCLES VERSUS K4

Abstract

Given three graphs G, H and K we write K → (G, H), if in any red/blue coloring of the edges of K there exists a red copy of G or a blue copy of H. The Ramsey number r(G, H) is defined as the smallest natural number n such that Kn → (G, H) and the star-critical Ramsey number r∗(G, H) is defined as the smallest positive integer k such that Kn−1 ⊔ K1,k → (G, H), where n is the Ramsey number r(G, H). When n ≥ 3, we show that r∗(Cn, K4) = 2n except for r∗(C3, K4) = 8 and r∗(C4, K4) = 9. We also characterize all Ramsey critical r(Cn, K4) graphs.