# Why is Adleman's molecular algorithm for Hamiltonian Path linear?

In Adleman's 1994 paper (archived), he describes a method of manipulating DNA molecules in a lab that results in a solution to the Hamiltonian Path problem with high probability.

He claims that "The number of different oligonucleotides required should grow linearly with the number of edges", and this makes sense given the molecular encoding of the edges described in the paper, but he also claims that "the number of procedures required should grow linearly with the number of vertices in the graph". Why is this last claim true? In other words, why isn't the assembly of edge molecules counted towards this calculation? If it were, the complexity of lab procedures would be $$O(|V|^2)$$, as the number of edges in a graph is bound by this function.

Thank you.