  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.
