Online

Mattern, Amelia

Mathematics

2015

MS

In this paper we extend the study of Heffter arrays and the biembedding of graphs on orientable surfaces first discussed by Archdeacon in 2014. We begin with the definitions of Heffter systems, Heffter arrays, and their relationship to orientable biembeddings through current graphs. We then focus on two specific cases. We first prove the existence of embeddings for every K₆n+1) with every edge on a face of size 3 and a face of size n. We next present partial results for biembedding K₁₀n+1 with every edge on a face of size 5 and a face of size n. Finally, we address the more general question of ordering subsets of Zn {0}. We conclude with some open conjectures and further explorations.