On relative simple Heffter spaces
Laura Johnson, Lorenzo Mella, Anita Pasotti
TL;DR
This work introduces relative Heffter spaces as a natural generalization of Heffter arrays and Heffter spaces, focusing on resolvable configurations whose points form a half-set of $G\setminus J$ and whose blocks sum to zero in $G$. It constructs two infinite families of simple cyclic relative Heffter spaces, analyzes their density, and derives strong consequences such as globally simple relative Heffter arrays and dense Heffter nets, including maximal or near-maximal densities for prime and prime-power parameters. The authors further show how these spaces yield sets of mutually orthogonal cyclic cycle decompositions of complete multipartite graphs and establish biembeddings into orientable surfaces via compatible orderings and the Crazy Knight's Tour problem. They also provide explicit globally simple cyclically diagonal arrays and discuss direct connections to biembeddings and toroidal/path-like tours, enriching the combinatorial design toolbox with new dense, structured constructions. Overall, the paper advances the theory of relative Heffter structures and their geometric and graph-theoretic applications.
Abstract
In this paper, we introduce the concept of a relative Heffter space which simultaneously generalizes those of relative Heffter arrays and Heffter spaces. Given a subgroup $J$ of an abelian group $G$, a relative Heffter space is a resolvable configuration whose points form a half-set of $G\setminus{J}$ and whose blocks are all zero-sum in $G$. Here we present two infinite families of relative Heffter spaces satisfying the additional condition of being simple. As a consequence, we get new results on globally simple relative Heffter arrays, on mutually orthogonal cycle decompositions and on biembeddings of cyclic cycle decompositions of the complete multipartite graph into an orientable surface.