A survey of Heffter arrays

Abstract

Heffter arrays were introduced by Archdeacon in 2015 as an interesting link between combinatorial designs and topological graph theory. Since the initial paper on this topic, there has been a good deal of interest in Heffter arrays as well as in related topics such as the sequencing of subsets of a group, biembeddings of cycle systems on a surface, and orthogonal cycle systems. This survey presents an overview of the current state of the art of this topic. We begin with an introduction to Heffter arrays for the reader who is unfamiliar with the subject, then we give a unified and comprehensive presentation of the major results, showing some proof methods also. This survey also includes sections on the connections of Heffter arrays to several other combinatorial objects, such as problems on partial sums and sequenceability, biembedding graphs on surfaces, difference families and orthogonal graph decompositions. These sections are followed by a section discussing the variants and generalizations of Heffter arrays which have been proposed. The survey itself is complemented by a list of unsolved problems as well as an updated and complete bibliography.