2026/04/30 by Erik Pelttari, Selda Küçükçifçi, E. Şule Yazıcı
Mathematics · #math.CO #msc:05B30 #msc:05C51
29 pages, 14 tables. This research was funded by TÜBİTAK, Grant/Award Number: 124F360
arxiv created 2026/07/29 · arxiv updated 2026/07/30
Heffter arrays are combinatorial structures used to construct orthogonal cyclic cycle decompositions and biembeddings of complete graphs onto surfaces. A Heffter array H(m,n;h,k) is an m × n partially filled array with distinct nonzero entries from ℤ2nk+1 such that each row contains h filled cells, each column contains k filled cells, the elements in the filled cells form a half-set of ℤ2nk+1, and every row and column sums to zero modulo 2nk+1. If these row and column sums equal zero over the integers, the structure is called an integer Heffter array. Furthermore, such an array is called globally simple if the partial sums of the entries in each row and column, evaluated in their natural order, are distinct modulo 2nk+1. When m=n and h=k, the array is square and denoted by H(n;k). While the existence of globally simple square Heffter arrays has been established for several congruence classes, the cases where k ≡ 1,2 \pmod4 for k > 10 have remained an open problem [1]. In this work, we address this gap in the literature by explicitly constructing globally simple integer Heffter arrays H(n;k) for the previously open cases where k ≡ 1 \pmod4 and n ≡ 0,3 \pmod4. Consequently, these constructions guarantee the existence of orthogonal cyclic k-cycle decompositions of the complete graph K2nk+1 for these parameters. [1] J.H. Dinitz and A. Pasotti. A survey of Heffter arrays. In C.J. Colbourn, editor, New Advances in Designs, Codes and Cryptography, volume 86, pages 353-392. Springer Nature Switzerland, 2024.