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Plongements polyédraux tendus et nombre chromatique relatif des surfaces à bord

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.
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https://hal.archives-ouvertes.fr/hal-01082198
Contributor : Pierre Jammes <>
Submitted on : Wednesday, November 12, 2014 - 8:24:04 PM
Last modification on : Monday, October 12, 2020 - 2:28:05 PM
Long-term archiving on: : Friday, February 13, 2015 - 11:36:57 AM

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  • HAL Id : hal-01082198, version 1

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Pierre Jammes. Plongements polyédraux tendus et nombre chromatique relatif des surfaces à bord. 2014. ⟨hal-01082198⟩

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