On the compatibility of a system of random comparisons

We consider a random graph whose each cycle is marked with a certain probability depending on the cycle length. The asymptotic behaviour of the probability of absence of non-marked cycles is described. As a corollary, the asymptotic behaviour of the probability of consistency of a system of random congruences modulo two with random non-equiprobable right-hand sides and also of a system with non-random right-hand sides is described.