28 research outputs found
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
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 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
Composition rings from formal power series rings
We study the ideals in some Dickson nearrings of polynomials and formal power series. For some of their related quotients, we introduce variants and generalizations, and construct composition rings also