On The Existence Of Almost Resolvable Cycle Systems With Odd Order

A k-cycle system of order n is a pair (V, C), where C is a collection of A;-cycles which partition the edges of Kn with vertex set V. B. Alspach et al. have proved that a k-cyde system of order n exists if and only if 3≤k≤n,n =1 (mod 2) and n(n - 1) = 0 (mod 2k).A resolvable k-cycle system of order n is a k-cycle system (V, C), where C can be partitioned into 2-factors. It has been proved by B. Alspach et al. that a resolvable k-cycle system of order n exists if and only if 3≤k≤n,n =1 (mod 2), and n = 0 (mod k).In a k-cycle system of order n, a collection of (n-1)/k disjoint k-cycles is called an almost parallel class, and a collection of (n-1)/2k disjoint k-cycles is called a half-parallel class. A k-cyde system of order n whose cyde set can be partitioned into (n - 1)/2 almost parallel classes and a half-parallel dass is called an almost resolvable k-cycle system, denoted by k-ARCS(n).S. A. Vanstone et al. started the research of the existence of an almost re-solvable k-cyde system. Many authors contributed to the following results. For k ∈ {3,4,5,6, 7,8,9,10,14}, t ≥ 1 and n = 2kt+1, there exists a k-ARCS(n) except for(k,n) ∈ {(3,7), (3,13), (4,9)} and except possibly for (k,n) e {(8,33), (14,57)}. For t > 1, t≠2,3,5,11≤k≤49 and k = 1 (mod 2), there is a k-ARCS(n).In this thesis, we almost solve the existence of a fc-ARCS(n), and prove that for any t > 1 and odd k≥11, there exists a k-ARCS(2kt+1) except possibly for t = 2.