  We show that the expected time for a random walk on a (multi-)graph $G$ to
traverse all $m$ edges of $G$, and return to its starting point, is at most
$2m^2$; if each edge must be traversed in both directions, the bound is $3m^2$.
Both bounds are tight and may be applied to graphs with arbitrary edge lengths,
with implications for Brownian motion on a finite or infinite network of total
edge-length $m$.
