Relative Heffter arrays and biembeddings

Marco Antonio Pellegrini, Simone Costa, Anita Pasotti

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

Abstract

[Autom. eng. transl.] Relative Heffter arrays, denoted by Ht (m, n; s, k), have been introduced as a generalization of the classical concept of Heffter array. A Ht (m, n; s, k) is an m × n partially filled array with elements in ℤv, 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 ∈ ℤv 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 (mod 4) and for k = 3 with t = n, 2n, any odd n .
Original languageEnglish
Pages (from-to)241-271
Number of pages31
JournalArs Mathematica Contemporanea
Volume18
DOIs
Publication statusPublished - 2020

Keywords

  • Heffter array
  • biembedding
  • multipartite complete graph

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