Decompositions of Complete Multigraphs into Cyclic Designs

Let and be positive integer, denote a complete multigraph. A decomposition of a graph is a set of subgraphs of whose edge sets partition the edge set of . The aim of this paper, is to decompose a complete multigraph into cyclic -cycle system according to specified conditions. As the main consequence, construction of decomposition of into cyclic Hamiltonian wheel system, where , is also given. The difference set method is used to construct the desired designs.

Heffter Arrays and Biembedding Graphs on Surfaces

A Heffter array is an $m \times n$ matrix with nonzero entries from $\mathbb{Z}_{2mn+1}$ such that i) every row and column sum to 0, and ii) exactly one of each pair $\{x,-x\}$ of nonzero elements appears in the array. We construct some Heffter arrays. These arrays are used to build current graphs used in topological graph theory. In turn, the current graphs are used to embed the complete graph $K_{2mn+1}$ so that the faces can be 2-colored, called a biembedding. Under certain conditions each color class forms a cycle system. These generalize biembeddings of Steiner triple systems. We discuss some variations including Heffter arrays with empty cells, embeddings on nonorientable surfaces, complete multigraphs, and using integer arithmetic in place of modular arithmetic.