Relative Heffter arrays and biembeddings - Costa, Simone (Ars mathematica contemporanea, 2020)
Relative Heffter arrays, denoted by ?$H_t(m,n; s,k)$?, have been introduced as a generalization of the classical concept of Heffter array. A ?$H_t(m,n; s,k)$? is an ?$m× n$? partially filled array with elements in ?$\mathbb{Z}_v$?, where ?$v=2nk+t$?, whose rows contain ?$s$? filled cells and whose columns contain ?$k$? filled cells, such that the elements in every row and column sum to zero and, for every ?$x\in \mathbb{Z}_v$? not belonging to the subgroup of order ?$t$?, either ?$x$? or ?$-x$? appears in the array. In this paper we show how relative Heffter arrays can be used to construct biembeddings of cyclic cycle decompositions of the complete multipartite graph ?$K_{\frac{2nk+t}{t}× t}$? into an orientable surface. In particular, we construct such biembeddings providing integer globally simple square relative Heffter arrays for ?$t=k=3,5,7,9$? and ?$n\equiv 3 \pmod 4$? and for ?$k=3$? with ?$t=n,2n$?, any odd ?$n$?.
Keywords: biembedding; bivložitev; Heffter array; Heffterjeva matrika; multipartite complete graph; polni večdelni graf