Globally simple Heffter arrays and orthogonal cyclic cycle decompositions

TL;DR

This paper introduces globally simple Heffter arrays that enable the construction of orthogonal cyclic cycle decompositions of complete and cocktail party graphs, providing explicit examples for cycles up to length 10 and applications to surface embeddings.

Contribution

It defines globally simple Heffter arrays and demonstrates their use in creating cycle decompositions and biembeddings, advancing combinatorial design theory.

Findings

Explicit constructions for cycle lengths up to 10

Orthogonal cyclic cycle decompositions of complete graphs

Biembeddings of cycle decompositions on surfaces

Abstract

In this paper we introduce a particular class of Heffter arrays, called globally simple Heffter arrays, whose existence gives at once orthogonal cyclic cycle decompositions of the complete graph and of the cocktail party graph. In particular we provide explicit constructions of such decompositions for cycles of length $k\leq 10$. Furthermore, starting from our Heffter arrays we also obtain biembeddings of two $k$-cycle decompositions on orientable surfaces.