Edge Disjoint Hamilton Cycles
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چکیده
In the late 70s, it was shown by Komlós and Szemerédi ([7]) that for p = lnn+ln lnn+c n , the limit probability for G(n, p) to contain a Hamilton cycle equals the limit probability for G(n, p) to have minimum degree at least 2. A few years later, Ajtai, Komlós and Szemerédi ([1]) have shown a hitting time version of this in the G(n,m) model. Say a graph G has property H if it contains bδ(G)/2c edge disjoint Hamilton cycles, plus a further edge disjoint near perfect matching in the case δ(G) is odd. Is it true that for every 0 ≤ p ≤ 1 the random graph G(n, p) has property H with high probability? This is clear whenever δ(G) = 0. In the early 80s, Bollobás and Frieze ([3]) have proved that conjecture for δ(G) = O(1). In this talk I plan to prove the result for p(n) ≤ (1 + o(1)) lnn/n. This is a result of Frieze and Krivelevich from ’08 ([4]). Remark 1. The conjecture is nowadays known to be true for every p. It was proved for the range ln n/n ≤ p ≤ 1 − ln n/n1/4 by Knox, Kühn and Osthus in ’13 ([6]), in a rather technically complicated paper. Later, Krivelevich and Samotij ([8]) have closed the gap for the sparse case, and Kühn and Osthus ([9]) have closed the gap for the dense case.
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