Van der Holst and Pendavingh introduced a graph parameter σ, which coincides with the more famous Colin de Verdière graph parameter µ for small values. However, the definition of σ is much more geometric/topological directly reflecting embeddability properties of the graph. They proved µ(G) ≤ σ(G) + 2 and conjectured µ(G) ≤ σ(G) for any graph G. We confirm this conjecture. As far as we know, this is the first topological upper bound on µ(G) which is, in general, tight. Equality between µ and σ does not hold in general as van der Holst and Pendavingh showed that there is a graph G with µ(G) ≤ 18 and σ(G) ≥ 20. We show that the gap appears on much smaller values, namely, we exhibit a graph H for which µ(H) ≤ 7 and σ(H) ≥ 8. We also prove that, in general, the gap can be large: The incidence graphs Hq of finite projective planes of order q satisfy µ(Hq) ϵ O(q3/2) and σ(Hq) ≥ q2.

The classical theorem of Fáry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fáry’s theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding . The main question addressed in this paper is whether given a closed surface S , there exists a Riemannian metric for which every topologically embeddable graph admits a shortest path embedding. This question is also motivated by various problems regarding crossing numbers on surfaces. We observe that the round metrics on the sphere and the projective plane have this property. We provide flat metrics on the torus and the Klein bottle which also have this property. Then we show that for the unit square flat metric on the Klein bottle there exists a graph without shortest path embeddings. We show, moreover, that for large g , there exist graphs G embeddable into the orientable surface of genus g , such that with large probability a random hyperbolic metric does not admit a shortest path embedding of G , where the probability measure is proportional to the Weil–Petersson volume on moduli space. Finally, we construct a hyperbolic metric on every orientable surface S of genus g , such that every graph embeddable into S can be embedded so that every edge is a concatenation of at most O ( g ) shortest paths.