TY - JOUR

T1 - On two Hamilton cycle problems in random graphs

AU - Frieze, Alan

AU - Krivelevich, Michael

N1 - Funding Information:
∗ Research supported in part by NSF grant CCR-0200945. ∗∗Research supported in part by USA-Israel BSF Grant 2002-133 526/05 from the Israel Science Foundation. Received July 26, 2006 and in revised form December 4, 2006

PY - 2008/8

Y1 - 2008/8

N2 - We study two problems related to the existence of Hamilton cycles in random graphs. The first question relates to the number of edge disjoint Hamilton cycles that the random graph Gn,pcontains. δ(G)/2 is an upper bound and we show that if p ≤ (1 + o(1)) ln n/n then this upper bound is tight whp. The second question relates to how many edges can be adversarially removed from Gn,p without destroying Hamiltonicity. We show that if p ≥ K ln n/n then there exists a constant α > 0 such that whp G - H is Hamiltonian for all choices of H as an n-vertex graph with maximum degree Δ(H) ≤ αK ln n.

AB - We study two problems related to the existence of Hamilton cycles in random graphs. The first question relates to the number of edge disjoint Hamilton cycles that the random graph Gn,pcontains. δ(G)/2 is an upper bound and we show that if p ≤ (1 + o(1)) ln n/n then this upper bound is tight whp. The second question relates to how many edges can be adversarially removed from Gn,p without destroying Hamiltonicity. We show that if p ≥ K ln n/n then there exists a constant α > 0 such that whp G - H is Hamiltonian for all choices of H as an n-vertex graph with maximum degree Δ(H) ≤ αK ln n.

UR - http://www.scopus.com/inward/record.url?scp=58449122859&partnerID=8YFLogxK

U2 - 10.1007/s11856-008-1028-8

DO - 10.1007/s11856-008-1028-8

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AN - SCOPUS:58449122859

VL - 166

SP - 221

EP - 234

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

ER -