Maps on graphs can be deformed to be coincidence free

Abstract

We give a construction to remove coincidence points of continuous maps on graphs ($1$-complexes) by changing the maps by homotopies. When the codomain is not homeomorphic to the circle, we show that any pair of maps can be changed by homotopies to be coincidence free. This means that there can be no nontrivial coincidence index, Nielsen coincidence number, or coincidence Reidemeister trace in this setting, and the results of our previous paper ``A formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles'' are invalid.