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Experimental Proof for Symmetric Ramsey Numbers

Journal of The Korea Society of Computer and Information
Abbr : JKSCI
2015, 20(3), pp.69-74
Publisher : The Korean Society Of Computer And Information
Research Area : Computer Science

1강릉원주대학교

• ABSTRACT

This paper offers solutions to unresolved 43 ≤ R(5,5) ≤ 49and 102 ≤ R(6,6) ≤ 165 problems ofRamsey’s number. The Ramsey’s number R(s,t) of a complete graph Kn dictates that n-1 number ofincidental edges of a arbitrary vertex v is dichotomized into two colors:(n-1)/2=R and (n-1)/2=B . Therefore, if one introduces the concept of distance to the vertex , one may construct a partite graph Kn = KL + v + KR , to satisfy (n-1)/2=R of {KL,v} and (n-1)/2=B of {v,KR}. Subsequently, given that KL forms the color R of Ks-1 , Ks is attainable. Likewise, given that KR forms the color B of Kt-1 , Kt is obtained. By following the above-mentioned steps, R(s,t) = Kn was obtained, satisfying necessary andsufficient conditions where, for KL and KR , the maximum distance should be even and incidental edges ofall vertices should be equal are satisfied. This paper accordingly proves R(5,5)=43 and R(6,6) = 91.
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