56 research outputs found
Square Integer Heffter Arrays with Empty Cells
A Heffter array is an matrix with nonzero entries
from such that each row contains filled cells and
each column contains filled cells, every row and column sum to 0, and
no element from appears twice. Heffter arrays are useful in
embedding the complete graph on an orientable surface where the
embedding has the property that each edge borders exactly one cycle and one
cycle. Archdeacon, Boothby and Dinitz proved that these arrays can be
constructed in the case when , i.e. every cell is filled. In this paper we
concentrate on square arrays with empty cells where every row sum and every
column sum is in . We solve most of the instances of this case.Comment: 20 pages, including 2 figure
Globally simple Heffter arrays and orthogonal cyclic cycle decompositions
In this paper we introduce a particular class of Heffter arrays, called
globally simple Heffter arrays, whose existence gives at once orthogonal cyclic
cycle decompositions of the complete graph and of the cocktail party graph. In
particular we provide explicit constructions of such decompositions for cycles
of length . Furthermore, starting from our Heffter arrays we also
obtain biembeddings of two -cycle decompositions on orientable surfaces.Comment: The present version also considers the problem of biembedding
A problem on partial sums in abelian groups
In this paper we propose a conjecture concerning partial sums of an arbitrary
finite subset of an abelian group, that naturally arises investigating simple
Heffter systems. Then, we show its connection with related open problems and we
present some results about the validity of these conjectures
A generalization of Heffter arrays
In this paper we define a new class of partially filled arrays, called
relative Heffter arrays, that are a generalization of the Heffter arrays
introduced by Archdeacon in 2015. Let be a positive integer, where
divides , and let be the subgroup of of order .
A Heffter array over relative to is an
partially filled array with elements in such that:
(a) each row contains filled cells and each column contains filled
cells; (b) for every , either or appears
in the array; (c) the elements in every row and column sum to . Here we
study the existence of square integer (i.e. with entries chosen in
and
where the sums are zero in ) relative Heffter arrays for ,
denoted by . In particular, we prove that for , with
, there exists an integer if and only if one of the
following holds: (a) is odd and ; (b) and is even; (c) . Also, we show how these arrays give
rise to cyclic cycle decompositions of the complete multipartite graph
Relative Heffter arrays and biembeddings
Relative Heffter arrays, denoted by , have been
introduced as a generalization of the classical concept of Heffter array. A
is an partially filled array with elements
in , where , whose rows contain filled cells and
whose columns contain filled cells, such that the elements in every row and
column sum to zero and, for every not belonging to the
subgroup of order , either or 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
into an orientable surface. In particular, we
construct such biembeddings providing integer globally simple square relative
Heffter arrays for and and for with
, any odd .Comment: arXiv admin note: text overlap with arXiv:1906.0393
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