2021/05/26 by Matthew de Courcy-Ireland, de Courcy-Ireland, Matthew · 2 citations
Mathematics · #05C10 #05C50 #11D25 #11F72 #37P25 #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2105.12411
openalex publication_date 2021/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.