1 research outputs found
Biembeddings of cycle systems using integer Heffter arrays
In this paper we will show the existence of a face -colourable biembedding
of the complete graph onto an orientable surface where each face is a cycle of
a fixed length , for infinitely many values of . In particular, under
certain conditions, we show that there exists at least non-isomorphic face -colourable biembeddings of in which
all faces are cycles of length . These conditions are: , and either is prime or and
implies . To achieve this result we begin by verifying the
existence of non-equivalent Heffter arrays, , which
satisfy the conditions: (1) for each row and each column the sequential partial
sums determined by the natural ordering must be distinct modulo ; (2)
the composition of the natural orderings of the rows and columns is equivalent
to a single cycle permutation on the entries in the array. The existence of
Heffter arrays that satisfy condition (1) was established earlier in
\cite{BCDY} and in this current paper we vary this construction and show that
there are at least such non-equivalent that
satisfy condition (1) and then show that each of these Heffter arrays also
satisfy condition (2) under certain conditions.Comment: arXiv admin note: text overlap with arXiv:1906.0736