On Diophantine graphs

TL;DR

This paper investigates Diophantine graphs, finite graphs with vertices as positive integers linked by edges when their product plus one is a perfect square, providing bounds, properties, and extendability results.

Contribution

It introduces new results on the structure, bounds, and extendability of Diophantine graphs, expanding understanding of their combinatorial and number-theoretic properties.

Findings

Bounds for the maximum number of edges

Chromatic number estimates

Extendability properties of 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.