31 research outputs found

    A generalization of the problem of Mariusz Meszka

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    Mariusz Meszka has conjectured that given a prime p=2n+1 and a list L containing n positive integers not exceeding n there exists a near 1-factor in K_p whose list of edge-lengths is L. In this paper we propose a generalization of this problem to the case in which p is an odd integer not necessarily prime. In particular, we give a necessary condition for the existence of such a near 1-factor for any odd integer p. We show that this condition is also sufficient for any list L whose underlying set S has size 1, 2, or n. Then we prove that the conjecture is true if S={1,2,t} for any positive integer t not coprime with the order p of the complete graph. Also, we give partial results when t and p are coprime. Finally, we present a complete solution for t<12.Comment: 15 page

    A problem on partial sums in abelian groups

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    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

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    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≤10k\leq 10. Furthermore, starting from our Heffter arrays we also obtain biembeddings of two kk-cycle decompositions on orientable surfaces.Comment: The present version also considers the problem of biembedding

    Relative Heffter arrays and biembeddings

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    Relative Heffter arrays, denoted by Ht(m,n;s,k)\mathrm{H}_t(m,n; s,k), have been introduced as a generalization of the classical concept of Heffter array. A Ht(m,n;s,k)\mathrm{H}_t(m,n; s,k) is an m×nm\times n partially filled array with elements in Zv\mathbb{Z}_v, where v=2nk+tv=2nk+t, whose rows contain ss filled cells and whose columns contain kk filled cells, such that the elements in every row and column sum to zero and, for every x∈Zvx\in \mathbb{Z}_v not belonging to the subgroup of order tt, either xx or −x-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 K2nk+tt×tK_{\frac{2nk+t}{t}\times t} into an orientable surface. In particular, we construct such biembeddings providing integer globally simple square relative Heffter arrays for t=k=3,5,7,9t=k=3,5,7,9 and n≡3(mod4)n\equiv 3 \pmod 4 and for k=3k=3 with t=n,2nt=n,2n, any odd nn.Comment: arXiv admin note: text overlap with arXiv:1906.0393

    A generalization of Heffter arrays

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    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 v=2nk+tv=2nk+t be a positive integer, where tt divides 2nk2nk, and let JJ be the subgroup of Zv\mathbb{Z}_v of order tt. A Ht(m,n;s,k)H_t(m,n; s,k) Heffter array over Zv\mathbb{Z}_v relative to JJ is an m×nm\times n partially filled array with elements in Zv\mathbb{Z}_v such that: (a) each row contains ss filled cells and each column contains kk filled cells; (b) for every x∈Zv∖Jx\in \mathbb{Z}_v\setminus J, either xx or −x-x appears in the array; (c) the elements in every row and column sum to 00. Here we study the existence of square integer (i.e. with entries chosen in ±{1,…,⌊2nk+t2⌋}\pm\left\{1,\dots,\left\lfloor \frac{2nk+t}{2}\right\rfloor \right\} and where the sums are zero in Z\mathbb{Z}) relative Heffter arrays for t=kt=k, denoted by Hk(n;k)H_k(n;k). In particular, we prove that for 3≤k≤n3\leq k\leq n, with k≠5k\neq 5, there exists an integer Hk(n;k)H_k(n;k) if and only if one of the following holds: (a) kk is odd and n≡0,3(mod4)n\equiv 0,3\pmod 4; (b) k≡2(mod4)k\equiv 2\pmod 4 and nn is even; (c) k≡0(mod4)k\equiv 0\pmod 4. Also, we show how these arrays give rise to cyclic cycle decompositions of the complete multipartite graph
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