Which sequences of adjacent edges of a polyhedron could be considered to be a geodesic? The edges of a face most surely will not, but the "equator" of the octahedron eventually will. But for what reasons? How do the defining property of a geodesic - having zero geodesic curvature - apply to a sequence of edges?
(One crude guess: any sequence of edges that pairwise don't share a face? What does this have to do with curvature?)