On Almost Resolvable Cycle Systems

A(k,λ)-cycle system of order v is a pair(v,C),where C is a collection of k-cycles which partition the edges of λKv with vertex set V.Further,if C can be partitioned into 2-factors,then(V,C)is called a resolvable.Let(V,C)be a(k,λ)-cycle system of order v,where v = kq + r and 0<r<k.A collection of m pairwise vertex-disjoint k-cycles of Kv is called an almost parallel class if m = q,or a short parallel class if 0<m<q.If C can be partitioned into[λCv2/v-r]almost parallel classes and a short parallel class which consists of[λCv2-[λcv2/v-r]·k/ki]k-cycles of Kv,then(V,C)is called an almost resolvable(k,λ)-cycle system,denoted by(k,λ)-ARCS(v).The almost resolvable(k,λ)-cycle system is a generalized form of a resolvable(k,λ)-cycle system.It was first defined and investigated by S.A.Vanstone et al.in 1993.Since then,many scholars have contributed a lot of research on this issue,and the main results are mainly focus on the case λ = r = 1.There are also some results on λ>1 and r>1,but the problem is far from being solved.In this thesis we study the existence of a almost resolvable(k,A)-cycle system for k ＝4,6,12 by using algebraic methods and combinatorial methods.We use some group structure to construct almost resolvable(k,λ)-cycle systems with special au-tomorphism in our direct constructions.For recursive constructions,we mainly use"filling in holes" constructions with cycle frames and incomplete almost resolvable cycle systems as input designs.The main results of this thesis are:(1)For k ∈ {4,6} and all A ≥ 1,we prove the necessary and sufficient condition for the existence of a(k,λ)-ARCS(v).(2)We prove the sufficient and necessary condition for the existence of a(12,1)-ARCS(v)with three possible exceptions.