  Given a proper edge-coloring of a loopless multigraph, the palette of a
vertex is defined as the set of colors of the edges which are incident with it.
The palette index of a multigraph is defined as the minimum number of distinct
palettes occurring among the vertices, taken over all proper edge-colorings of
the multigraph itself. In this framework, the palette multigraph of an
edge-colored multigraph is defined in this paper and some of its properties are
investigated. We show that these properties can be applied in a natural way in
order to produce the first known family of multigraphs whose palette index is
expressed in terms of the maximum degree by a quadratic polynomial. We also
attempt an analysis of our result in connection with some related questions.
