Publication

Non-planarity of Markoff graphs mod p

Matthew De Courcy-Ireland
2024
Journal Articles

Abstract

We prove the non-planarity of a family of 3-regular graphs constructed from the solutions to the Markoff equation x2 + y2 + z2 = 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.

Official source

https://infoscience.epfl.ch/entities/publication/fcc355c9-493c-4700-b78f-2f3e3ce5a47d

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Non-planarity of Markoff graphs mod p

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