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On Diophantine graphs

Gergő Batta, Lajos Hajdu, András Pongrácz

Abstract

Diophantine tuples are of ancient and modern interest, with a huge literature. In this paper we study Diophantine graphs, i.e., finite graphs whose vertices are distinct positive integers, and two vertices are linked by an edge if and only if their product increased by one is a square. We provide various results for Diophantine graphs, including extendability properties, lower- and upper bounds for the maximum number of edges and chromatic numbers.

On Diophantine graphs

Abstract

Diophantine tuples are of ancient and modern interest, with a huge literature. In this paper we study Diophantine graphs, i.e., finite graphs whose vertices are distinct positive integers, and two vertices are linked by an edge if and only if their product increased by one is a square. We provide various results for Diophantine graphs, including extendability properties, lower- and upper bounds for the maximum number of edges and chromatic numbers.

Paper Structure

This paper contains 12 sections, 18 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Extendability of Diophantine graphs
  4. Bounds for the number of edges in Diophantine graphs
  5. Chromatic number of Diophantine graphs
  6. Proof of the results on extensions of Diophantine graphs
  7. Proof of the results concerning the number of edges of Diophantine graphs
  8. Proof of the lower bound
  9. Proof of the upper bounds
  10. Proof of the results related to chromatic numbers of Diophantine graphs
  11. Remarks and open problems
  12. Acknowledgments

Key Result

Theorem 2.1

Let $V=\{v_1,\dots,v_n\}\subseteq \mathbb{N}$. Then each of the following conditions is satisfied by infinitely many positive integers $w$ whose square-free part differs from that of $v_k$ for all $1\leq k\leq n$:

Theorems & Definitions (32)

  • Theorem 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 22 more