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Multiplicité du spectre de Steklov sur les surfaces et nombre chromatique

We prove several results about the multiplicity of the first Steklov eigenvalues on compact surfaces with boundary. We improve some bounds on the multiplicity, especially for the first eigenvalue, and we prove they are sharp on some surfaces of small genus. In a previous article, we defined a new chromatic invariant of surfaces with boundary and conjectured that this invariant is related to the bound on the first eigenvalue. In the present article, we study this invariant, and prove that the conjecture is true when the known bound is sharp.