Cycle systems in the complete bipartite graph minus a one-factor

AbstractLet Kn,n−I denote the complete bipartite graph with n vertices in each part from which a 1-factor I has been removed. An m-cycle system of Kn,n−I is a collection of m-cycles whose edges partition Kn,n−I. Necessary conditions for the existence of such an m-cycle system are that m⩾4 is even, n⩾3 is odd, m⩽2n, and m|n(n−1). In this paper, we show these necessary conditions are sufficient except possibly in the case that m≡0(mod4) with n<m<2n

Block Disjoint Difference Families for Steiner Triple Systems: v &amp;equiv; 3 mod 6

A block disjoint (v,k,) difference family is a difference family with disjoint blocks. We show that disjoint (v; 3; 1) difference families exist for all v j 3 mod 6 with v 3. 1 Introduction Let G be a group of order v. A family of k-tuples of elements from G is a (v,k,) difference family if the collection of orbits of the k-tuples (disregarding repeated k-tuples) under the action of G form a balanced incomplete block design, BIBD(v; k; ). If the k-tuples are pairwise disjoint, call the family a block disjoint (v,k,) difference family. In this paper we show that there exists a block disjoint (v,3,1) difference family for all v j 3 mod 6. This is a companion paper to [2] which considered the case of block disjoint (v,3,1) difference family for v j 1 mod 6. The reader is refered to that paper for background information. 2 Constructions In this section we give constructions for block disjoint (v,3,1) difference families when v j 3 mod 6. We give a different construction for each congru..

The existence of square non-integer Heffter arrays

A Heffter array H(n; k) is an n x n matrix such that each row and column contains k filled cells, each row and column sum is divisible by 2nk + 1 and either x or -x appears in the array for each integer