Heffter arrays and related topics

Heffter arrays are a class of combinatorial arrays introduced by Archdeacon in 2015 as an interesting link between Combinatorial Designs and Topological Graph Theory. In fact, they are a useful tool to construct cycle decompositions of complete graphs and their embeddings over orientable surfaces. More explicitly, an Heffter array is a partially filled matrix, whose entries form an half-set of a cyclic group, such that any two rows (respectively, any two columns) have the same number of filled cells, and every row and every column has sum equal to zero in the group. Since their definition, many variants and generalizations have been extensively studied given their various applications. The naming of this class of arrays is due to Heffter’s first difference problem, proposed in 1896 and eventually solved by Peltesohn more than forty years later, that guaranteed a complete solution for the existence of cyclic Steiner triple systems. It can then be proven that an Heffter array with certain parameters provide a generalized solution to Heffter’s difference problem. In the PhD thesis we give an overview of Heffter arrays, together with their applications to graph decompositions and embeddings. Then, we present some results that have been obtained during the course of the PhD program, that mainly concern weak Heffter arrays, non-zero sum Heffter arrays, and the construction of embeddings having a high degree of symmetry. We then consider the recently introduced notion of an Heffter space, strongly related to many well-known concepts of Finite Geometry, and we conclude by constructing an infinite class of Heffter spaces.