TY - JOUR

T1 - Cores of random graphs are born Hamiltonian

AU - Krivelevich, Michael

AU - Lubetzky, Eyal

AU - Sudakov, Benny

N1 - Funding Information:
Michael Krivelevich was supported in part by USA-Israel BSF Grant 2010115 and by grants 1063/08, 912/12 from Israel Science Foundation. Benny Sudakov was supported in part by NSF grant DMS-1101185, USA-Israeli BSF grant and by the Swiss National Science Foundation grant 200021-149111.

PY - 2014/7

Y1 - 2014/7

N2 - Let (Gt) be the random graph process (G0) is edgeless and Gt is obtained by adding a uniformly distributed new edge to Gt-1), and let tk 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 = tk its size jumps from 0 to linear in the number of vertices with high probability (w.h.p.). It is believed that for any k ≥3, the core is Hamiltonian upon creation w.h.p., and Bollob́as, Cooper, Fenner and Frieze further conjectured that it in fact admits ≥(k - 1)/2 edgedisjoint 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 ≥tk, that is, immediately as the k-core appears and indefinitely afterwards. Moreover, we prove that for large enough fixed k the k-core contains ≥(k - 3)/2 edge-disjoint Hamilton cycles w.h.p. for all t ≥tk.

AB - Let (Gt) be the random graph process (G0) is edgeless and Gt is obtained by adding a uniformly distributed new edge to Gt-1), and let tk 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 = tk its size jumps from 0 to linear in the number of vertices with high probability (w.h.p.). It is believed that for any k ≥3, the core is Hamiltonian upon creation w.h.p., and Bollob́as, Cooper, Fenner and Frieze further conjectured that it in fact admits ≥(k - 1)/2 edgedisjoint 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 ≥tk, that is, immediately as the k-core appears and indefinitely afterwards. Moreover, we prove that for large enough fixed k the k-core contains ≥(k - 3)/2 edge-disjoint Hamilton cycles w.h.p. for all t ≥tk.

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U2 - 10.1112/plms/pdu003

DO - 10.1112/plms/pdu003

M3 - Article

AN - SCOPUS:84904337738

SN - 0024-6115

VL - 109

SP - 161

EP - 188

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

IS - 1

ER -