Non-planarity of Markoff graphs mod p

TL;DR

This paper proves that certain graphs derived from solutions to the Markoff equation modulo large primes are non-planar, using topological and combinatorial methods, with explicit constructions for specific prime classes.

Contribution

It establishes the non-planarity of Markoff graphs mod p for large primes and specific residue classes, using Euler characteristic and cycle enumeration techniques.

Findings

Markoff graphs are non-planar for primes > 7

Explicit constructions confirm non-planarity for certain prime classes

Non-planarity linked to spectral gap assumptions

Abstract

We prove the non-planarity of a family of 3-regular graphs constructed from the solutions to the Markoff equation $x^2+y^2+z^2=xyz$ modulo prime numbers greater than 7. The proof uses Euler characteristic and an enumeration of the short cycles in these graphs. Non-planarity for large primes would follow assuming a spectral gap, which was the original motivation. For primes congruent to 1 modulo 4, or congruent to 1, 2, or 4 modulo 7, explicit constructions give an alternate proof of non-planarity.