Tight Heffter Arrays Exist for all Possible Values |
| |
| Authors: | Dan S Archdeacon Tomas Boothby Jeffrey H Dinitz |
| |
| Institution: | 1. Department of Mathematics and Statistics, University of Vermont, Burlington, Vermont;2. Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada |
| |
| Abstract: | A tight Heffter array  is an  matrix with nonzero entries from  such that (i) the sum of the elements in each row and each column is 0, and (ii) no element from  appears twice. We prove that  exist if and only if both m and n are at least 3. If H has the property that all entries are integers of magnitude at most  , every row and column sum is 0 over the integers, and H also satisfies  ), we call H an integer Heffter array. We show integer Heffter arrays exist if and only if  . Finally, an integer Heffter array is shiftable if each row and column contains the same number of positive and negative integers. We show that shiftable integer arrays exists exactly when both  are even. |
| |
| Keywords: | Heffter arrays biembedding Skolem sequences Steiner triple systems |
|
|
is an 
matrix with nonzero entries from 
such that (i) the sum of the elements in each row and each column is 0, and (ii) no element from 
appears twice. We prove that 
exist if and only if both m and n are at least 3. If H has the property that all entries are integers of magnitude at most 
, every row and column sum is 0 over the integers, and H also satisfies 
), we call H an integer Heffter array. We show integer Heffter arrays exist if and only if 
. Finally, an integer Heffter array is shiftable if each row and column contains the same number of positive and negative integers. We show that shiftable integer arrays exists exactly when both 
are even.