BORSUK–ULAM PROPERTY FOR GRAPHS

Abstract

For finite connected graphs Γ and 𝐺, with Γ admitting a free involution 𝜏, we characterize the based homotopy classes 𝛼 ∈ [Γ, 𝐺] for which the Borsuk–Ulam property holds in the sense of Gonçalves, Guaschi, and Casteluber Laass, i.e., the homotopy classes 𝛼 such that each of the representatives 𝑓 ∈ 𝛼 satisfies 𝑓(𝑥) = 𝑓(𝜏 · 𝑥) for some 𝑥 ∈ Γ. This is attained through a graph-braid-group perspective aided by the use of discrete Morse theory.