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The Markoff graph modulo $p$ is known to be connected for all but finitelymany primes $p$ (see Eddy, Fuchs, Litman, Martin, Tripeny, and Vanyo[arxiv:2308.07579]), and it is conjectured that these graphs are connected forall primes. In this paper, we provide an algorithmic realization of the processintroduced by Bourgain, Gamburd, and Sarnak [arxiv:1607.01530] to test whethera Markoff graph modulo $p$ is connected for arbitrary primes. Our algorithmruns in $o(p^{1 + \epsilon})$ time for every $\epsilon > 0$. We demonstratethis algorithm by confirming that the Markoff graph modulo $p$ is connected forall primes less than one million.

An almost linear time algorithm testing whether the Markoff graph modulo $p$ is connected