On the trace of random walks on random graphs

We study graph-theoretic properties of the trace of a random walk on a random graph. We show that for any $\varepsilon>0$ there exists $C>1$ such that the trace of the simple random walk of length $(1+\varepsilon)n\ln{n}$ on the random graph $G\sim G(n,p)$ for $p>C\ln{n}/n$ is, with high probability, Hamiltonian and $\Theta(\ln{n})$-connected. In the special case $p=1$ (i.e. when $G=K_n$), 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 $k$'th time, the trace becomes $2k$-connected.

[1]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[2]  Alan M. Frieze,et al.  Component structure of the vacant set induced by a random walk on a random graph , 2011, SODA '11.

[3]  Uriel Feige,et al.  A Tight Lower Bound on the Cover Time for Random Walks on Graphs , 1995, Random Struct. Algorithms.

[4]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[5]  A. Frieze,et al.  Introduction to Random Graphs , 2016 .

[6]  B. Sudakov,et al.  Pseudo-random Graphs , 2005, math/0503745.

[7]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[8]  J. Komlos,et al.  First Occurrence of Hamilton Cycles in Random Graphs , 1985 .

[9]  Recurrence of random walk traces , 2006, math/0603060.

[10]  Mark Jerrum,et al.  Approximating the Permanent , 1989, SIAM J. Comput..

[11]  Béla Bollobás,et al.  Random Graphs: Notation , 2001 .

[12]  Svante Janson,et al.  Random graphs , 2000, Wiley-Interscience series in discrete mathematics and optimization.

[13]  L. Pósa,et al.  Hamiltonian circuits in random graphs , 1976, Discret. Math..

[14]  A. Rbnyi.  ON THE EVOLUTION OF RANDOM GRAPHS , 2001 .

[15]  Alan M. Frieze,et al.  The cover time of sparse random graphs. , 2003, SODA '03.

[16]  Michael Krivelevich,et al.  Hamilton cycles in highly connected and expanding graphs , 2006, Comb..

[17]  Michael Krivelevich,et al.  Hitting time results for Maker‐Breaker games , 2010, Random Struct. Algorithms.

[18]  Augusto Teixeira,et al.  GIANT VACANT COMPONENT LEFT BY A RANDOM WALK IN A RANDOM d-REGULAR GRAPH , 2010, 1012.5117.

[19]  W. T. Gowers,et al.  RANDOM GRAPHS (Wiley Interscience Series in Discrete Mathematics and Optimization) , 2001 .

[20]  B. Bollobás,et al.  Random Graphs of Small Order , 1985 .

[21]  P. Erd6s.  ON A CLASSICAL PROBLEM OF PROBABILITY THEORY b , 2001 .

[22]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[23]  Ben Barber,et al.  Random Walks on Quasirandom Graphs , 2013, Electron. J. Comb..

[24]  László Lovász,et al.  Random Walks on Graphs: A Survey , 1993 .