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The relative chromatic number $c\_0(S)$ of a compact surface $S$ withboundary is defined as the supremum of the chromatic numbers of graphs embeddedin $S$ with all vertices on $\partial S$. This topological invariant wasintroduced 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 oftight polyhedral embeddings of $S$ byproving two results. The first one is thatif there is a tight polyhedral embedding of $S$ in $\R^n$ which is notcontained in a hyperplane, then $n\leq c\_0(S)-1$. The second result is thatthis inequality is sharp for surfaces of small genus.

Tight polyhedral embeddings and relative chromatic number of surfaces with boundary