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arXiv · 2105.12411

Non-planarity of Markoff graphs mod p

Matthew de Courcy-Ireland
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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.

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Matthew de Courcy-Ireland. 2021-05-26. Non-planarity of Markoff graphs mod p. https://arxiv.org/abs/2105.12411

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