Borsuk-Ulam property for graphs

Abstract

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 Gonçalves, 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.

11 pages, 2 figures