An Infinite Family of Counterexamples to a Conjecture on Distance Magic Labeling

Abstract

This work is about a partition problem which is an instance of the distance magic graph labeling problem. Given positive integers n, k and p 1 ≤ p 2 ≤ ⋯ ≤ p k such that p 1 + ⋯ + p k = n and k divides ∑ i = 1 n i , we study the problem of characterizing the cases where it is possible to find a partition of the set { 1 , 2 , … , n } into k subsets of respective sizes p 1 , … , p k , such that the element sum in each subset is equal. Using a computerized search we found examples showing that the necessary condition, ∑ i = 1 p 1 + ⋯ + p j ( n − i + 1 ) ≥ j ( n + 1 2 ) / k for all j = 1 , … , k , is not generally sufficient, refuting a past conjecture. Moreover, we show that there are infinitely many such counter-examples. The question whether there is a simple characterization is left open and for all we know the corresponding decision problem might be NP-complete.