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Maximizing the number of integer pairs summing to powers of 2 via graph labeling and solving restricted systems of linear (in)equations

Max A. Alekseyev

TL;DR

This work addresses the problem of maximizing the number of 2-sums among pairs in an $n$-element set by recasting it as labeling a graph with pairwise distinct integers so each edge sum is a power of $2$. It introduces a computational framework that reduces graph admissibility to solving a matrix equation $M L = X$ and then solving a derived system of equations and inequations in powers of $2$ via the SolveInPowers algorithm, embedded in GraphSolve. Using these methods, the authors determine $g(n)$ for $n\leq 21$, proving $g(n)=\ell(n)$ for $n$ in $[12,21]$, and enumerate all minimal forbidden subgraphs up to order $11$ and all maximum admissible graphs up to order $20$, including exact counts. The work provides detailed algorithmic tools, complexity insights, and open-source code to study power-of-two graph labelings and related extremal questions, with implications for squarefree graphs and combinatorial labeling problems.

Abstract

We address the problem of finding sets of integers of a given size with a maximum number of pairs summing to powers of $2$. By fixing particular pairs, this problem reduces to finding a labeling of the vertices of a given graph with pairwise distinct integers such that the endpoint labels for each edge sum up to a power of $2$. We propose an efficient algorithm for this problem, which at its core relies on another algorithm that, given two sets of linear homogeneous polynomials with integer coefficients, computes all variable assignments to powers of $2$ that nullify polynomials from the first set but not from the second. With the proposed algorithms, we determine the maximum size of graphs of order $n$ that admit such a labeling for all $n\leq 21$, and construct the maximum admissible graphs for $n\leq 20$. We also identify the minimal forbidden subgraphs of order $\leq 11$, whose presence prevents the graphs from having such a labeling.

Maximizing the number of integer pairs summing to powers of 2 via graph labeling and solving restricted systems of linear (in)equations

TL;DR

This work addresses the problem of maximizing the number of 2-sums among pairs in an -element set by recasting it as labeling a graph with pairwise distinct integers so each edge sum is a power of . It introduces a computational framework that reduces graph admissibility to solving a matrix equation and then solving a derived system of equations and inequations in powers of via the SolveInPowers algorithm, embedded in GraphSolve. Using these methods, the authors determine for , proving for in , and enumerate all minimal forbidden subgraphs up to order and all maximum admissible graphs up to order , including exact counts. The work provides detailed algorithmic tools, complexity insights, and open-source code to study power-of-two graph labelings and related extremal questions, with implications for squarefree graphs and combinatorial labeling problems.

Abstract

We address the problem of finding sets of integers of a given size with a maximum number of pairs summing to powers of . By fixing particular pairs, this problem reduces to finding a labeling of the vertices of a given graph with pairwise distinct integers such that the endpoint labels for each edge sum up to a power of . We propose an efficient algorithm for this problem, which at its core relies on another algorithm that, given two sets of linear homogeneous polynomials with integer coefficients, computes all variable assignments to powers of that nullify polynomials from the first set but not from the second. With the proposed algorithms, we determine the maximum size of graphs of order that admit such a labeling for all , and construct the maximum admissible graphs for . We also identify the minimal forbidden subgraphs of order , whose presence prevents the graphs from having such a labeling.
Paper Structure (7 sections, 4 theorems, 4 equations, 4 figures, 3 tables, 3 algorithms)

This paper contains 7 sections, 4 theorems, 4 equations, 4 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Algorithm for testing graph admissibility
  3. Solving a system of (in)equations in powers of $2$
  4. Minimal forbidden subgraphs
  5. Computing values of $g(n)$
  6. Maximum admissible graphs
  7. Discussion

Key Result

Theorem 1

Let $v$ be an integer column vector of size $k\geq 0$, and $A$ be a $k\times s$ integer matrix with pairwise distinct rows. Then there exists an integer linear combination of the columns of $A$ such that adding it to $v$ results in a vector with pairwise distinct elements.

Figures (4)

  • Figure 1: Minimal forbidden subgraphs of order $7$.
  • Figure 2: Minimal forbidden subgraphs of order $10$.
  • Figure 3: Maximum admissible graphs of order $14$ with the corresponding vertex labelings.
  • Figure 4: Maximum admissible graphs of order $16$ with the corresponding vertex labelings.

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4
  • proof