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We give a construction to remove coincidence points of continuous maps ongraphs (1-complexes) by changing the maps by homotopies. When the codomain isnot homeomorphic to the circle, we show that any pair of maps can be changed byhomotopies to be coincidence free. This means that there can be no nontrivialcoincidence index, Nielsen coincidence number, or coincidence Reidemeistertrace in this setting, and the results of our previous paper "A formula for thecoincidence Reidemeister trace of selfmaps on bouquets of circles" are invalid.

Maps on graphs can be deformed to be coincidence-free