Tight polyhedral embeddings and relative chromatic number of surfaces with boundary

TL;DR

This paper explores the relationship between the relative chromatic number of a surface with boundary and its tight polyhedral embeddings, establishing bounds and sharpness for small genus surfaces.

Contribution

It demonstrates that the relative chromatic number bounds the dimension of tight polyhedral embeddings and proves the bound's sharpness for small genus surfaces.

Findings

If a tight polyhedral embedding exists, the embedding dimension is at most the relative chromatic number minus one.

The established inequality between embedding dimension and relative chromatic number is sharp for small genus surfaces.

The study links topological invariants with geometric embedding properties of surfaces.

Abstract

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.