C.J. Jayawardene, D. Narváez, S.P. Radziszowski
Star-critical Ramsey numbers for cycles versus K4
Discussiones Mathematicae Graph Theory
Received 25.09.2017, Revised 05.11.2018, Accepted 05.11.2018, doi: 10.7151/dmgt.2190
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 \sqcup 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.
Ramsey theory, star-critical Ramsey numbers