  The relative chromatic number $c\_0(S)$ of a compact surface $S$ with
boundary is defined as the supremum of the chromatic numbers of graphs embedded
in $S$ with all vertices on $\partial S$. This topological invariant was
introduced for the study of the multiplicity of the first Steklov eigenvalue of
$S$. In this article, we show that $c\_0(S)$ is also relevant for the study of
tight polyhedral embeddings of $S$ byproving two results. The first one is that
if there is a tight polyhedral embedding of $S$ in $\R^n$ which is not
contained in a hyperplane, then $n\leq c\_0(S)-1$. The second result is that
this inequality is sharp for surfaces of small genus.
