TY - JOUR

T1 - Relative Heffter arrays and biembeddings

AU - Pellegrini, Marco Antonio

AU - Costa, Simone

AU - Pasotti, Anita

PY - 2020

Y1 - 2020

N2 - [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 .

AB - [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 .

KW - Heffter array

KW - biembedding

KW - multipartite complete graph

KW - Heffter array

KW - biembedding

KW - multipartite complete graph

UR - http://hdl.handle.net/10807/162777

U2 - 10.26493/1855-3974.2110.6f2

DO - 10.26493/1855-3974.2110.6f2

M3 - Article

VL - 18

SP - 241

EP - 271

JO - Ars Mathematica Contemporanea

JF - Ars Mathematica Contemporanea

SN - 1855-3966

ER -