Further Results On Fine Structures Of Three Latin Squares

Let L_k = (a_ij^k)_v×v, k＝1, 2, 3 be Latin squares on symbol set S containingv elements. DefineC_l＝{(a_ij¹,a_aj²,a_aj³)∶|{a_ij¹,a_aj²,a_aj³}|＝l,i,j＝1,2,…,v}.The fine structure of A Latin squares L₁, L₂,…,L_λis the vector (t, s), where t is thenumber of elements in C₂ and s is the number of elements in C₂∪C₃. Let Fin(v) denotethe set of all integer pairs (t, s) for which there exist three Latin squares of order v withfine structure (t, s).Fin(v) with v≥9 is completely determined in^[2]. In this thesis, Fin(8) is finallydetermined, and some results on Fin(v) with v = 5, 6, 7 are also updated.The first chapter introduces the basic definition and some known results. In thesecond chapter, we present some new lemmas. Then, we give the proof of (1, 18) (?)Fin(8). At last, we give more non-existence examples and the proof of the (2, 14)(?)Fin(v) with v = 5, 6, 7. In the third chapter, we summarize the results obtained in thethesis.