A Note on Star Critical Ramsey (Cn,K6)  Numbers for Large n

Jayawardene, Chula J.; Navaratna, W. C. W.

doi:10.22457/apam.621v20n1a2

Cited by 2 publications

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“…Given two graphs G and H, we say that Kn → (G, H), if any red and blue two colouring of Kn contains a copy of G in red or a copy of H in bue. Studies on Ramsey numbers/Multipartite Ramsey numbers related to different classes have been studied extensively in the past few decades (see [4][5][6][7]9]). Studies on Star-critical Ramsey numbers related to different classes of graphs are trees vs complete graphs [3], paths vs. paths [2], stars vs. stripes [1] and complete graphs vs stripes are some such examples (also see [8,10]).…”

Section: Introductionmentioning

confidence: 99%

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

Jayawardene¹,

Senadheera²,

Fernando³

et al. 2020

APAM

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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.

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