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Globally simple Heffter arrays and orthogonal cyclic cycle decompositions
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In this paper, a particular class of simple Heffter arrays, called globally simple heffter array, whose existence gives at once orthogonal cyclic cycle decompositions of the complete graph and of the cocktail party graph was introduced.Abstract:
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 $k\leq 10$. Furthermore, starting from our Heffter arrays we also obtain biembeddings of two $k$-cycle decompositions on orientable surfaces.read more
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A problem on partial sums in abelian groups
TL;DR: This paper proposes a conjecture concerning partial sums of an arbitrary finite subset of an abelian group, that naturally arises investigating simple Heffter systems and shows its connection with related open problems and presents some results about the validity of these conjectures.
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A generalization of Heffter arrays
TL;DR: In this article, the authors define a new class of partially filled arrays, called relative Heffter arrays, which are a generalization of the Heffler arrays introduced by Archdeacon in 2015.
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A problem on partial sums in abelian groups
TL;DR: In this article, a conjecture concerning partial sums of an arbitrary finite subset of an abelian group, that naturally arises investigating simple Heffter systems, is proposed and its connection with related open problems is discussed.
Posted Content
Relative Heffter arrays and biembeddings
TL;DR: It is shown how relative Heffter arrays can be used to construct biembeddings of cyclic cycle decompositions of the complete multipartite graph K_{\frac{2nk+t}{t}\times t}$ into an orientable surface.
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The existence of square non-integer Heffter arrays
TL;DR: In this paper, Dinitz and Wanless showed that such an integer Heffter array can be found whenever the column sum is divisible by a factor of 2nk+1.
References
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On Orthogonal Arrays
Esther Seiden,Rita Zemach +1 more
TL;DR: In this paper, it was shown that one can construct orthogonal arrays with the maximum number of constraints (i.e., k + n + 1, 2, t + 1) provided that k = 2.
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Latin Squares: New Developments in the Theory and Applications
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Existence of cyclic k -cycle systems of the complete graph
Marco Buratti,Alberto Del Fra +1 more
TL;DR: A cyclic p-cycle system of Kv with p being a prime exists for all admissible values of v but (p,v) ≠ (3,9), which was previously known only for p = 3, 5, 7.
Journal ArticleDOI
Orientable embedding of Cayley graphs
TL;DR: In this paper, the transition probabilities pij(t) (i^j) of a Markov chain were examined, and it was shown that the transition probability of a chain is a function of the regenerative property of the state 0.
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Cyclic Hamiltonian cycle systems of the complete graph
Marco Buratti,Alberto Del Fra +1 more
TL;DR: It is proved that there exists a cyclic Hamiltonian k-cycle system of the complete graph if and only if k is odd but k≠15 and pα with p prime and α>1.