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Two different problems are studied: For an infinite locally finite connected graph $G$, let $p_c(G)$ be the critical value for the existence of an infinite cluster in iid bond percolation on $G$ and let $P_c = \sup\{p_c(G): G \text{ transitive }, p_c(G) q$ implies that for any monotone coupling $(X',Y')$ of $X$ and $Y$ the edges for which $X'$ and $Y'$ disagree form infinite connected component(s) with positive probability? Let $p_d(G)$ be the infimum of such $q$'s (including $q=1$) and let $P_d = \sup\{p_d(G): G \text{ transitive }, p_c(G) p_c (G)$? It is shown that if one restricts attention to biregular planar graphs then these two problems can be treated in a similar way and all the above questions are positively answered. We also give examples to show that if one drops the assumption of transitivity, then the answer to the above two questions is no. Furthermore it is shown that for any bounded-degree bipartite graph $G$ with $p_c(G) 1$.

On Disagreement Percolation and Maximality of the Critical Value for iid Percolation