  For finite connected graphs $\Gamma$ and $G$, with $\Gamma$ admitting a free
involution $\tau$, we characterize the based homotopy classes
$\alpha\in[\Gamma,G]$ for which the Borsuk-Ulam property holds in the sense of
Gon\c{c}alves, Guaschi and Casteluber-Laass, i.e., the homotopy classes
$\alpha$ so that each of its representatives $f\in\alpha$ satisfies $f(x) =
f(\tau\cdot x)$ for some $x\in\Gamma$. This is attained through a
graph-braid-group perspective aided by the use of discrete Morse theory.
