TY - JOUR

T1 - On the trace of random walks on random graphs

AU - Frieze, Alan

AU - Krivelevich, Michael

AU - Michaeli, Peleg

AU - Peled, Ron

N1 - Publisher Copyright:
© 2017 London Mathematical Society

PY - 2018/4

Y1 - 2018/4

N2 - We study graph-theoretic properties of the trace of a random walk on a random graph. We show that for any ε > 0 there exists C > 0 such that the trace of the simple random walk of length (1 + ε)n ln n on the random graph G ∼ G(n, p) for p> C ln n/n is, with high probability, Hamiltonian and Θ(ln n)-connected. In the special case p = 1 (that is, when G = Kn), we show a hitting time result according to which, with high probability, exactly one step after the last vertex has been visited, the trace becomes Hamiltonian, and one step after the last vertex has been visited for the Kth time, the trace becomes 2k-connected.

AB - We study graph-theoretic properties of the trace of a random walk on a random graph. We show that for any ε > 0 there exists C > 0 such that the trace of the simple random walk of length (1 + ε)n ln n on the random graph G ∼ G(n, p) for p> C ln n/n is, with high probability, Hamiltonian and Θ(ln n)-connected. In the special case p = 1 (that is, when G = Kn), we show a hitting time result according to which, with high probability, exactly one step after the last vertex has been visited, the trace becomes Hamiltonian, and one step after the last vertex has been visited for the Kth time, the trace becomes 2k-connected.

KW - 05C45

KW - 05C80

KW - 05C81 (primary)

KW - 60G50 (secondary)

UR - http://www.scopus.com/inward/record.url?scp=85036570711&partnerID=8YFLogxK

U2 - 10.1112/plms.12083

DO - 10.1112/plms.12083

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AN - SCOPUS:85036570711

VL - 116

SP - 847

EP - 877

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

SN - 0024-6115

IS - 4

ER -