The generalized almost resolvable cycle system problem

Abstract

Let (X, C) be a k-cycle system of order n, with vertex set X (of cardinality n) and collection of k-cycles C. Suppose n=kq+r where r<k. An almost parallel class of C is a collection of q=(n−r)/k pairwise vertex-disjoint k-cycles of C. Each almost parallel class thus will miss r of the n vertices in X. The k-cycle system (X,C) is said to be almost resolvable if C can be partitioned into almost parallel classes such that the remaining k-cycles are vertex disjoint. (These remaining k-cycles are referred to as a short parallel class.)

In this paper we (a) construct an almost resolvable 10-cycle system of every order congruent to 5 (mod 20), and (b) construct an almost resolvable 10-cycle system of order 41, thus completing the result that the spectrum for almost resolvable 10-cycle systems consists of all orders congruent to 1 or 5 (mod 20).