Start a random walk from each vertex of a graph $G$. Let the walkers evolve independently, except that when two of the walkers meet (ie. occupy the same vertex at the same time), they coalesce into one single walker. (Alternatively, one may label the walkers with numbers $1,\dots,n$ and say that walker $i$ is killed at the first time it meets a walker with smaller index.)

It is not hard to show that, if $G$ is connected and finite, there will be a first time $C$ when all walks will have coalesced into one ($C$ is the first time when only one alive particle remains in the "killing description"). The following is Open Problem $13$ in Chapter $14$ of Aldous and Fill's manuscript: http://www.stat.berkeley.edu/~aldous/RWG/book.html.

**Problem:** Prove that there exists a universal constant such that:
$$E(C)\leq K\max_{v,w\in V(G)}E_v(H_w)$$
where $V(G)$ is the vertex set of $G$ and $H_w$ is the hitting time of vertex $w$.

My **question** is: does anyone know of any work on this problem? I know there is a paper by Cox which studies the distributional limit of $C$ over $Z_n^d$ for $n\gg 1$ and $d$ fixed; a solution for the Problem follows from this in this specific family of graphs. What else is out there? Has the problem actually been solved by someone?