Abstract. 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.

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 + t be a positive integer, where t divides 2nk, and let J be the subgroup of Zv of order t. A Ht(m, n; s, k) Heffter array over Zv relative to J is an m × n partially filled array with elements in Zv such that: (i) each row contains s filled cells and each column contains k filled cells; (ii) for every x ∈ Z 2nk+t \ J, either x or −x appears in the array; (iii) the elements in every row and column sum to 0. In particular, here we study the existence for t = k of integer (i.e. the entries are chosen in ± 1, . . . , 2nk+t 2 and the sums are zero in Z) square relative Heffter arrays.

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 Kp 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 ≤ 11.2010 Mathematics Subject Classification. 05C38.

The conjecture of Peter Horak and Alex Rosa (generalizing that of Marco Buratti) states that a multiset L of ν-1 positive integers not exceeding ⌊ν/2⌋ is the list of edge-lengths of a suitable Hamiltonian path of the complete graph with vertex-set {0,1,⋯,ν-1} if and only if the following condition (here reformulated in a slightly easier form) is satisfied: for every divisor d of ν, the number of multiples of d appearing in L is at most v-d. In this paper we do some preliminary discussions on the conjecture, including its relationship with graph decompositions. Then we prove, as main result, that the conjecture is true whenever all the elements of L are in {1,2,3,5

In this paper we define a new class of partially filled arrays, called λ-fold relative Heffter arrays, that are a generalisation of the Heffter arrays introduced by Archdeacon in 2015. After showing the connection of this new concept with several other ones, such as signed magic arrays, graph decompositions and relative difference families, we determine some necessary conditions and we present existence results for infinite classes of these arrays. In the last part of the paper we also show that these arrays give rise to biembeddings of multigraphs into orientable surfaces and we provide infinite families of such biembeddings. To conclude, we present a result concerning pairs of λ-fold relative Heffter arrays and covering surfaces.