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An infinite family of counterexamples to a conjecture on distance magic labeling

Ehab Ebrahem, Shlomo Hoory, Dani Kotlar

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 $p_1\le p_2\le \cdots\le p_k$ such that $p_1+\cdots+p_k=n$ and $k$ divides $\sum_{i=1}^ni$, we study the problem of characterizing the cases where it is possible to find a partition of the set $\{1,2,\ldots,n\}$ into $k$ subsets of respective sizes $p_1,\dots,p_k$, such that the element sum in each subset is equal. Using a computerized search we found examples showing that the necessary condition, $\sum_{i=1}^{p_1+\cdots+p_j} (n-i+1)\ge j{\binom{n+1}{2}}/k$ for all $j=1,\ldots,k$, 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.

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 and such that and divides , we study the problem of characterizing the cases where it is possible to find a partition of the set into subsets of respective sizes , such that the element sum in each subset is equal. Using a computerized search we found examples showing that the necessary condition, for all , 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.
Paper Structure (9 sections, 6 theorems, 27 equations, 2 figures, 4 tables, 1 algorithm)

This paper contains 9 sections, 6 theorems, 27 equations, 2 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Definitions and Notation
  3. Constructing balanced partitions
  4. Initialization methods
  5. Reducing the value of $d(\mathcal{A})$ by transpositions
  6. Criteria for non-equitability
  7. Experimental results
  8. Infinite families of non-equitable SSC partitions
  9. Conclusion

Key Result

Theorem 4.2

Let $P = [2^e, p^f, \ldots] \in AP_{n,k}$ with $e,f>0$ and $p \ge 3$, with $C,c,h$ as defined in Notation notn:h, such that $f>h$, and Then $P$ is non-equitable.

Figures (2)

  • Figure 1: Number of ascending partitions by category as a function of $n$: non-cumulative (top), cumulative (bottom).
  • Figure 2: Scatter plot of n vs. k for all non-equitable SSC ascending partitions. The red dot size is proportional to the logarithm of the number of such partitions. The 6 lines depict the families described above.

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.5
  • Definition 2.7
  • Definition 2.8
  • Example 2.9
  • proof
  • Definition 2.10
  • Definition 3.1
  • Definition 3.3
  • ...and 15 more