### Coincidence of maps from two-complexes into graphs

#### Abstract

The main theorem of this article provides a necessary and sufficient condition for a pair of maps from a two-complex into a one-complex (a graph) can be homotoped to be coincidence free. As a consequence of it, we prove that a pair of maps from a two-complex into the circle can be homotoped to be coincidence free if and only if the two maps are homotopic. We also obtain an alternative proof for the known result that every pair of maps from a graph into the bouquet of a circle and an interval can be homotoped to be coincidence free. As applications of the main theorem, we characterize completely when a pair of maps from the bi-dimensional torus into the bouquet of a circle and an interval can be homotoped to be coincidence free, and we prove that every pair of maps from the Klein bottle into such a bouquet can be homotoped to be coincidence free.

#### Keywords

Coincidence theory; two-complex; graphs

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