TY - JOUR
T1 - Relative Heffter arrays and biembeddings
AU - Costa, Simone
AU - Pasotti, Anita
AU - Pellegrini, Marco Antonio
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
SN - 1855-3966
VL - 18
SP - 241
EP - 271
JO - Ars Mathematica Contemporanea
JF - Ars Mathematica Contemporanea
ER -