  In the two-type Richardson model on a graph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$, each vertex is at a given time in
state $0$, $1$ or $2$. A $0$ flips to a $1$ (resp.\ $2$) at rate $\lambda_1$
($\lambda_2$) times the number of neighboring $1$'s ($2$'s), while $1$'s and
$2$'s never flip. When $\mathcal{G}$ is infinite, the main question is whether,
starting from a single $1$ and a single $2$, with positive probability we will
see both types of infection reach infinitely many sites. This has previously
been studied on the $d$-dimensional cubic lattice $\mathbb{Z}^d$, $d\geq 2$,
where the conjecture (on which a good deal of progress has been made) is that
such coexistence has positive probability if and only if $\lambda_1=\lambda_2$.
In the present paper examples are given of other graphs where the set of points
in the parameter space which admit such coexistence has a more surprising form.
In particular, there exist graphs exhibiting coexistence at some value of
$\frac{\lambda_1}{\lambda_2} \neq 1$ and non-coexistence when this ratio is
brought closer to $1$.
