Irreversible adsorption of diatomics on crystalline surfaces is sometimes modeled as random dimer filling of adjacent pairs of sites on a lattice. We note that this process can be implemented in two distinct ways: (i) randomly pick adjacent pairs of sites,  jj’, and fill  jj’ only if both are empty (horizontal transition state); or (ii) randomly pick a single site,  j, and if  j and at least one neighbor are empty, then fill  j and a randomly chosen empty neighbor (vertical transition state). Here it is instructive to consider processes which also include competitive random monomer filling of single sites. We find that although saturation (partial) coverages differ little between the models for pure dimer filling, there is a significant difference for comparable monomer and dimer filling rates. We present exact results for saturation coverage behavior for a linear lattice, and estimates for a square lattice. Ramifications for simple models of CO oxidation on surfaces are indicated.