On $\lambda$-fold relative Heffter arrays and biembedding multigraphs on surfaces

TL;DR

This paper introduces $7$-fold relative Heffter arrays, explores their connections to graph theory, and demonstrates their use in biembedding multigraphs on surfaces, providing new combinatorial and topological insights.

Contribution

It defines a new class of arrays, establishes their existence, and links them to graph embeddings, expanding the combinatorial framework for surface embeddings.

Findings

Established necessary conditions for $7$-fold relative Heffter arrays.

Constructed infinite classes of these arrays.

Demonstrated biembeddings of multigraphs into surfaces using these arrays.

Abstract

In this paper we define a new class of partially filled arrays, called $\lambda$-fold relative Heffter arrays, that are a generalisation of the Heffter arrays introduced by Archdeacon in 2015. After showing the connection of this new concept with several other ones, such as signed magic arrays, graph decompositions and relative difference families, we determine some necessary conditions and we present existence results for infinite classes of these arrays. In the last part of the paper we also show that these arrays give rise to biembeddings of multigraphs into orientable surfaces and we provide infinite families of such biembeddings. To conclude, we present a result concerning pairs of $\lambda$-fold relative Heffter arrays and covering surfaces.