Cores of random graphs are born Hamiltonian

Let {Gt}t≥0 be the random graph process (G0 is edgeless and Gt is obtained by adding a uniformly distributed new edge to Gt−1), and let τk denote the minimum time t such that the k-core of Gt (its unique maximal subgraph with minimum degree at least k) is nonempty. For any fixed k ≥ 3 the k-core is known to emerge via a discontinuous phase transition, where at time t = τk its size jumps from 0 to linear in the number of vertices with high probability. It is believed that for any k ≥ 3 the core is Hamiltonian upon creation w.h.p., and Bollobás, Cooper, Fenner and Frieze further conjectured that it in fact admits b(k−1)/2c edge-disjoint Hamilton cycles. However, even the asymptotic threshold for Hamiltonicity of the k-core in G(n, p) was unknown for any k. We show here that for any fixed k ≥ 15 the k-core of Gt is w.h.p. Hamiltonian for all t ≥ τk, i.e., immediately as the k-core appears and indefinitely afterwards. Moreover, we prove that for large enough fixed k the k-core contains b(k − 3)/2c edge-disjoint Hamilton cycles w.h.p. for all t ≥ τk.