Maps on graphs can be deformed to be coincidence-free

We give constructions which can remove coincidence points of mappings on bouquets of circles by changing the maps by homotopies. When the codomain is a bouquet of at least 2 circles, we show that any pair of maps can be changed by homotopies to be coincidence free. This allows us to demonstrate that there can be no function on bouquets of circles which satisfies the natural properties expected of a coincidence index: additivity, homotopy invariance, and agreement with the fixed point index for selfmaps. Consequently, the Nielsen coincidence number and coincidence Reidemeister trace are not well-defined and the results of our previous paper ``A formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles" are invalid.