# A linear bound for the Colin de Verdi\`ere parameter $\mu$ for graphs embedded on surfaces

## TL;DR

This paper proves a combinatorial upper bound for the Colin de Verdière parameter of graphs embedded on surfaces, relating it to the surface's Euler characteristic.

## Contribution

It offers a new, self-contained combinatorial proof establishing a linear bound on the Colin de Verdière parameter based on surface topology.

## Key findings

- The Colin de Verdière parameter is bounded above by 7 minus twice the Euler characteristic.
- The proof is combinatorial and self-contained, avoiding complex algebraic methods.
- The bound applies to all graphs embedded on any surface.

## Abstract

We provide a combinatorial and self-contained proof that for all graphs $G$ embedded on a surface $S$, the Colin de Verdi\`ere parameter $\mu(G)$ is upper bounded by $7-2\chi(S)$.