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If I remember corretly, the mentioned problem from Run Twice contest appeared at Petrozavodsk summer camp 2022. My solution is to check whether the input graph has a $$$4$$$-clique: a random graph should not have it.
Nice! This solution feels a bit borderline to me: the graph has 1% of all edges when m=5000, so roughly speaking the probability of having a 4-clique is C(1000,4)*0.01^6 ~= 1000^4/24/100^6 = 1/24, so it might or might not appear in the ~100 testcases, not all of which have m=5000.