Borsuk–Ulam Property for Graphs II: The $$\mathbb{Z}_{n}$$-Action

For a finite group $$H$$ and connected topological spaces $$X$$ and $$Y$$ such that $$X$$ is endowed with a free left $$H$$ -action $$\tau$$ , we provide a geometric condition in terms of the existence of a commutative diagram of spaces (arising from the triple $$(X,Y;\tau)$$ ) to decide whether the Borsuk–Ulam property holds for based homotopy classes $$\alpha\in[X,Y]_{0}$$ , as well as for free homotopy classes $$\alpha\in[X,Y]$$ . Here a homotopy class $$\alpha$$ is said to satisfy the Borsuk–Ulam property if, for each of its representatives $$f\in\alpha$$ , there exists an $$H$$ -orbit, where $$f$$ fails to be injective. Our geometric characterization is attained by constructing an $$H$$ -equivariant map from $$X$$ to the classical configuration space $$F_{|H|}(Y)$$ . We derive an algebraic condition from the geometric characterisation, and show that former one is in fact equivalent to the latter one when $$X$$ and $$Y$$ are aspherical. We then specialize to the 1-dimensional case, i.e., when $$X$$ is an arbitrary connected graph, $$H$$ is cyclic, and $$Y$$ is either a tree, a circle, or the connected graph $$S^{1}\vee I$$ with two vertices and two edges one of which is a loop and the other is a closed interval. The graph-braid-group ingredient in our characterizations is then effectively controlled through the use of discrete Morse theory.