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.