Abstract
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.
Highlights
We study a relation between design theory, graph theory, and maps on surfaces
The rotation ρ on the current graph G plays two independent roles in this construction: i) ρ generates a monofacial embedding, and ii) each local rotation ρv is simple with respect to the current assignment κ
We have introduced Heffter arrays and their relation with current graphs and with biembeddings
Summary
We study a relation between design theory, graph theory, and maps on surfaces. From design theory a Heffter system is used to construct a cyclic k-cycle system. The current graph with certain conditions is used to construct an orientable embedding of the complete graph K2mn+1 that is face 2-colorable, the boundaries of each color class forming a cycle system. Under some special conditions current graphs can be used to construct embeddings of a complete graph These embeddings were first used in the solution of the Map Color Theorem [20]. Theorem 1.1 Given a Heffter array H(m, n; s, t) with compatible orderings ωr on D(m, s) and ωc on D(n, t), there exists an orientable embedding of K2ms+1 such that every edge is on a face of size s and a face of size t.
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