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A generalization of Heffter arrays
Author(s) -
Costa Simone,
Morini Fiorenza,
Pasotti Anita,
Pellegrini Marco Antonio
Publication year - 2020
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21684
Subject(s) - combinatorics , mathematics , generalization , integer (computer science) , multipartite , order (exchange) , graph , discrete mathematics , column (typography) , computer science , mathematical analysis , physics , finance , quantum mechanics , quantum entanglement , economics , quantum , programming language , geometry , connection (principal bundle)
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 = 2 n k + t be a positive integer, where t divides 2 n k , and let J be the subgroup of Z v of order t . A H t ( m , n ; s , k ) Heffter array over Z v relative to J is an m × n partially filled array with elements in Z v such that (a) each row contains s filled cells and each column contains k filled cells; (b) for every x ∈ Z v \ J , either x or − x appears in the array; and (c) the elements in every row and column sum to 0 . Here we study the existence of square integer (i.e., with entries chosen in ± 1 , … ,2 n k + t 2and where the sums are zero in Z ) relative Heffter arrays for t = k , denoted by H k ( n ; k ) . In particular, we prove that for 3 ≤ k ≤ n , with k ≠ 5 , there exists an integer H k ( n ; k ) if and only if one of the following holds: (a) k is odd and n ≡ 0 , 3( mod 4 ) ; (b) k ≡ 2( mod 4 ) and n is even; (c) k ≡ 0( mod 4 ) . Also, we show how these arrays give rise to cyclic cycle decompositions of the complete multipartite graph.