We consider the combinatorial properties of the trace of a random walk on the complete graph and on the random graph $G(n,p)$. In particular, we study the appearance of a fixed subgraph in the trace. We prove that for a subgraph containing a cycle, the threshold for its appearance in the trace of a random walk of length $m$ is essentially equal to the threshold for its appearance in the random graph drawn from $G(n,m)$. In the case where the base graph is the complete graph, we show that a fixed forest appears in the trace typically much earlier than it appears in $G(n,m)$.
@article{10_37236_6169,
author = {Michael Krivelevich and Peleg Michaeli},
title = {Small subgraphs in the trace of a random walk},
journal = {The electronic journal of combinatorics},
year = {2017},
volume = {24},
number = {1},
doi = {10.37236/6169},
zbl = {1355.05232},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6169/}
}
TY - JOUR
AU - Michael Krivelevich
AU - Peleg Michaeli
TI - Small subgraphs in the trace of a random walk
JO - The electronic journal of combinatorics
PY - 2017
VL - 24
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/6169/
DO - 10.37236/6169
ID - 10_37236_6169
ER -
%0 Journal Article
%A Michael Krivelevich
%A Peleg Michaeli
%T Small subgraphs in the trace of a random walk
%J The electronic journal of combinatorics
%D 2017
%V 24
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/6169/
%R 10.37236/6169
%F 10_37236_6169
Michael Krivelevich; Peleg Michaeli. Small subgraphs in the trace of a random walk. The electronic journal of combinatorics, Tome 24 (2017) no. 1. doi: 10.37236/6169
