α-resolvable Cycle Systems For Length6

In this paper,we discuss the existence problem of the α-resolvable cycle systems for length6.Let λKv be a complete multigraph of order v and index λ. A cycle of length m is a sequence of m distinct vertices u1,u2,...,um,denoted by(u1,u2,...,um),and its edge set is {{ui,ui+1):i=1,2,...,m-1}∪u{{u1,um}}.If the edges of a λKv can be decomposed into some cycles of length m,then these cycles are called an m-cycle system,and denoted by m-CS(v,λ).A α-resolution class is a set of some cycles such that each point of λKv occurs in precisely α cycles.If an m-CS(v,λ)can be partitioned into some α-resolution classes,then it is called α-resolvable.The research about α-resolvable cycle systems began from the early90century.In1991,D.Jungnickel,R.C.Mullin,S.A.Vanstone and others gave the spectrum of α-resolvable cycle systems for length3.In1997,P. Gvozdjak gave the existence of resolvable m-CS(v)for length m≥3.In2008,Ma Xiuwen proved the existence of α-resolvable cycle systems for length4in his master’s thesis.In this paper,we use the method of direct constructions and recursive constructions to discuss the existence problem of α-resolvable6-CS(v,λ).We obtain the following results:For v≡0(mod6),v≡1(mod6),v≡4(mod6),v≡5(mod6),v≡8(mod12),v≡14(mod24),fhere exists a α-resolvable6-CS(v,λ)where α and λ satisfying the necessary conditions.