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Integral and rational graphs in the plane

Jozsef Solymosi

Abstract

We describe constructions of infinite graphs which are not representable as integral graphs in the plane, addressing a question of Erdős. We also mention some related problems.

Integral and rational graphs in the plane

Abstract

We describe constructions of infinite graphs which are not representable as integral graphs in the plane, addressing a question of Erdős. We also mention some related problems.
Paper Structure (8 sections, 1 theorem, 3 equations, 3 figures)

This paper contains 8 sections, 1 theorem, 3 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Non-integral graphs
  3. The first family of non-integral graphs
  4. The second family of non-integral graphs
  5. The third family of non-integral graphs
  6. Rational graphs
  7. Open problems
  8. Acknowledgements

Key Result

Theorem 1

If $P$ is an infinite point set in the plane, such that the distances are integers, $\overline{AB} \in \mathbb{Z}$ for any $A,B\in P$, then $P$ is a subset of a line.

Figures (3)

  • Figure 1: The line through $P$ and $Q$ intersects the $x$-axis between $F_1$ and $F_2$
  • Figure 2: With a fast-growing $|P_i|$ sequence this graph is not integral
  • Figure 3: With a fast-growing $\chi(G_i)$ sequence this graph is not integral

Theorems & Definitions (7)

  • Theorem 1: Anning and Erdős
  • proof
  • Claim 2
  • proof
  • Claim 3
  • proof
  • Definition 4