Abstract
We define a non-monotone digraph property TOUR1, a variant of the digraph property KERNEL, which refines the notion of maximal tournament. First we prove that there is a constant 0 < α < 1 such that TOUR1 is asymptotically almost surely true in random digraphs with constant arc probability p ≤ α and asymptotically almost surely false in random digraphs with constant arc probability p > α. Then we concentrate our study on random digraphs with arc probability close to a and we obtain a sharp threshold.
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