Tight polyhedral embeddings and relative chromatic number of surfaces with boundary

الأصل · EN

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 ∂ 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 ⁿ which is not contained in a hyperplane, then n≤ c_0(S)-1. The second result is that this inequality is sharp for surfaces of small genus.