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Let   $d\ge 3$   be a fixed integer. Let   $y:= y(p)$   be the probability that the root of an infinite   $d$  -regular tree belongs to an infinite cluster after   $p$  -bond-percolation. We show that for all constants   $b,\alpha&gt;0$   and   $1&lt;\lambda &lt; d-1$  , there exist constants   $c,C&gt;0$   such that the following holds. Let   $G$   be a   $d$  -regular graph on   $n$   vertices, satisfying that for every   $U\subseteq V(G)$   with   $|U|\le \frac{n}{2}$  ,   $e(U,U^{c})\ge b|U|$   and for every   $U\subseteq V(G)$   with   $|U|\le \log ^{C}n$  ,   $e(U)\le (1+c)|U|$  . Let   $p=\frac{\lambda }{d-1}$  . Then, with probability tending to one as   $n$   tends to infinity, the largest component   $L_{1}$   in the random subgraph   $G_{p}$   of   $G$   satisfies   $\left |1-\frac{|L_{1}|}{yn}\right |\le \alpha $  , and all the other components in   $G_{p}$   are of order   $O\left (\frac{\lambda \log n}{(\lambda -1)^{2}}\right )$  . This generalises (and improves upon) results for random   $d$  -regular graphs.

Components, Large and Small, Are as They Should Be II: Supercritical Percolation on Regular Graphs of Constant Degree