Relative Heffter arrays and biembeddings

2019/09/06 by Costa, Simone, Pasotti, Anita, Pellegrini, Marco Antonio · 2 citations
#05B20 #05B30 #05C10 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1909.03064

Abstract

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 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∈ 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_{(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≡ 3 \pmod 4 and for k=3 with t=n,2n, any odd n.

Cited by

• Globally Simple Heffter Arrays H(n;k) with k ≡ 1 \pmod4
• Further constructions of square integer relative Heffter arrays

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