Research On Planar Ramsey Numbers PR(K₄-e,K₆) And PR(C₄,K₇)

Ramsey theory is a cohesive subdiscipline of Discrete Mathematics, and graph Ramseytheory is an important branch of Ramsey theory. There are many interesting applications ofRamsey theory, these include results in number theory, algebra, geometry, topology, settheory, logic, ergodic theory, information theory and theoretical computer science. It isNP-hard to determine the value of Rarnsey number. Up to now, there are few Ramseynumbers which are known. In this dissertation, combining the computer constructive provewith mathematical prove, the planar Ramsey numbers are researched deeply.The definition of planar Ramsey number was firstly introduced by Walker in 1969.Bielak gave that PR(C₄, K₅)=13, PR(C₄, K₆)=17 and the lower bounds on PR(C₄, K_l).This article in these research's foundation, discovered that in the time which may tolerateare most with the critical chart strategy algorithm can prove only PR(K₄-e, K₅)=14 andconfirms PR(C₄, K₆)=17. In view of this algorithm this insufficiency, this article developedcomputation plane Ramsey to count PR(H₁, H₂) value new algorithm CPG. This algorithmhas not used the usual critical chart strategy, but aims in the plane Ramsey number flatnessthis limit, uses the non-horizontal plan to contain the homographism certainly in K₅ or K_{3, 3}sub graph this theorem, through increases the apex to construct since childhood with the sideapex all not including forbids sub graph H₁(H₁≌K₄-e或H₁≌C₄) n apex horizontal plan, thendetermines in these If has such chart, then charts whether to have chart G to cause H₂(?).Then PR(H₁, H₂)＞n+1, Otherwise, PR(H₁, H₂)≤n. This algorithm in Intel Intel P4 1.7G, on thememory 500M computer moved. The experiment indicated that this algorithm may confirmPR(K₄-e, K_l)(l≤6) and PR(C₄, K_l)(l≤7). the value, and might prove PR(K₄-e, K₆)=17 andPR(C₄, K₇)=20. Because side its algorithm does not exist deletes operates, the running ratealso enhanced more than 6 times compared to the critical chart strategy algorithm.Simultaneously this article applies the Euler's formula and three connection horizontal plan'splane inserting only nature the plane Ramsey number in the proof. Has proven strictly withthe PR(K₄-e, K₆)=17 and PR(C₄, K₇)=20. mathematical method brief proof PR(K₄-e, K₁)≥3l+(?)(l-1)/4(?)-2(l≥3) article gives back to following two suspicions: PR(C₄, K_l)=3l+(?)(l-1)/5(?)-2(l≥3) and PR(K₄-e, K_l)=3l+(?)(l-1)/4(?)-2(l≥3).