Papers › An almost linear time algorithm testing whether the Markoff graph modulo p is connected
An almost linear time algorithm testing whether the Markoff graph modulo p is connected
Colby Austin Brown
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The Markoff graph modulo p is known to be connected for all but finitely many primes p (see Eddy, Fuchs, Litman, Martin, Tripeny, and Vanyo [arxiv:2308.07579]), and it is conjectured that these graphs are connected for all primes. In this paper, we provide an algorithmic realization of the process introduced by Bourgain, Gamburd, and Sarnak [arxiv:1607.01530] to test whether the Markoff graph modulo p is connected for arbitrary primes. Our algorithm runs in o(p^(1 + ϵ)) time for every ϵ> 0. We demonstrate this algorithm by confirming that the Markoff graph modulo p is connected for all primes less than one million.
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