  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.
