Some New Results Of Super - Lap System

Let λKn denote the λ multipartite complete graph on n vertices in which each pair of vertices is joined by exactly λ edges. Let X be the vertex set of the graph G,|X|=n, if the edges of λKn can be partitioned into a set C of k-cycles, then (X,C) is called λ multipartite k;-cycle system with order n, denoted by (k,λ)-CS(n).Let (X, C) be a (k, A)-CS(n).(X, C) is called super-simple, denoted by (k,λ)-SCS(n), if any two distinct cycles C1, C2∈C have at most two points in common. Gronau and Muliin introduced the concept of super-simple in1992. The existence of a (4,2)-SCS(n) has been solved by Billington et al. in2011. Xiuwen Chen has proved that the necessary conditions for the existence of a (4,3)-SCS(n.) and (4,4)-SCS(n) are also sufficient.In this thesis, we shall use direct constructions and recursive constructions to prove the following results:(1) A (5,2)-SCS(n) exists if and only if n=0,1(mod5) and n≥15;(2) A (4,5)-SCS(n) exists if and only if n≥17and n=1(mod8) with a possible exception of n=17;(3) If n>6and n≠38, then there exists a (4,6)-SCS(4n-3).