On Star-critical (K1,n, K1,m + e) Ramsey Numbers

Abstract

We say that Kn → (G,H), if for every red/blue colouring of edges of the complete graph Kn, there exists a red copy of G, or a blue copy of H in the colouring of Kn. The Ramsey number r(G,H) is the smallest positive integer n such that Kn → (G,H). Let r(n,m)=r(Kn, Km). A closely related concept of Ramsey numbers is the Star-critical Ramsey number r*(G, H) defined as the largest value of k such that K r(G,H)-1 ˅ K 1,k → (G,H). Literature on survey papers in this area reveals many unsolved problems related to these numbers. One of these problems is the calculation of Ramsey numbers for certain classes of graphs. The primary objective of this paper is to calculate the Star critical Ramsey numbers for the case of Stars versus K1,m+e. The methodology that we follow in solving this problem is to first find a closed form for the Ramsey number r*(K1,n , K1,m+e) for all n, m ≥ 3. Based on the values of r*(K1,n , K1,m+e) for different n, m we arrive at a general formula for r*(K1,n , K1,m+e). Henceforth, we show that r*(K1,n , K1,m+e) = n+m-1 is defined by a piecewise function related to the three disjoint cases of n, m both even and n ≤ m - 2, n or m is odd and n ≤ m-2 and n > m-2.