  Let $(G_n)_{n \geq 1} = ((V_n,E_n))_{n \geq 1}$ be a sequence of finite,
connected, vertex-transitive graphs with volume tending to infinity. We say
that a sequence of parameters $(p_n)_{n \geq 1}$ in $[0,1]$ is supercritical
with respect to Bernoulli bond percolation $\mathbb P_p^G$ if there exists
$\varepsilon >0$ and $N<\infty$ such that \[
  \mathbb P_{(1-\varepsilon)p_n}^{G_n} \left( \text{the largest cluster
contains at least $\varepsilon |V_n|$ vertices}\right) \geq \varepsilon \] for
every $n\geq N$ with $p_n <1$. We prove that if $(G_n)_{n \geq 1}$ is sparse,
meaning that the degrees are sublinear in the number of vertices, then the
supercritical giant cluster is unique with high probability in the sense that
if $(p_n)_{n \geq 1}$ is supercritical then \[
  \lim_{n\to\infty}\mathbb P_{p_n}^{G_n} \left( \text{the second largest
cluster contains at least $c|V_n|$ vertices} \right) = 0 \] for every $c>0$.
This result is new even under the stronger hypothesis that $(G_n)_{n \geq 1}$
has uniformly bounded vertex degrees, in which case it verifies a conjecture of
Benjamini (2001). Previous work of many authors had established the same
theorem for complete graphs, tori, hypercubes, and bounded degree expander
graphs, each using methods that are highly specific to the examples they
treated. We also give a complete solution to the problem of supercritical
uniqueness for dense vertex-transitive graphs, establishing a simple necessary
and sufficient isoperimetric condition for uniqueness to hold.
