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Globally simple Heffter arrays H(n;k) when k≡0,3(mod4)
Discrete Mathematics ( IF 0.770 ) Pub Date : 2020-01-13 , DOI: 10.1016/j.disc.2019.111787
Kevin Burrage; Diane M. Donovan; Nicholas J. Cavenagh; Emine Ş. Yazıcı

Square Heffter arrays are n×n arrays such that each row and each column contains k filled cells, each row and column sum is divisible by 2nk+1 and either x or x appears in the array for each integer 1xnk. Archdeacon noted that a Heffter array, satisfying two additional conditions, yields a face 2-colourable embedding of the complete graph K2nk+1 on an orientable surface, where for each colour, the faces give a k-cycle system. Moreover, a cyclic permutation on the vertices acts as an automorphism of the embedding. These necessary conditions pertain to cyclic orderings of the entries in each row and each column of the Heffter array and are: (1) for each row and each column the sequential partial sums determined by the cyclic ordering must be distinct modulo 2nk+1; (2) the composition of the cyclic orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. We construct Heffter arrays that satisfy condition (1) whenever (a) k0(mod4); or (b) n1(mod4) and k3(mod4); or (c) n0(mod4), k3(mod4) and nk. As corollaries to the above we obtain pairs of orthogonal k-cycle decompositions of K2nk+1.

更新日期:2020-01-29

 

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