  Given an edge-coloring of a graph, the palette of a vertex is defined as the
set of colors of the edges which are incident with it. We define the palette
index of a graph as the minimum number of distinct palettes, taken over all
edge-colorings, occurring among the vertices of the graph. Several results
about the palette index of some specific classes of graphs are known. In this
paper we propose a different approach that leads to new and more general
results on the palette index. Our main theorem gives a sufficient condition for
a graph to have palette index larger than its minimum degree. In the second
part of the paper, by using such a result, we answer to two open problems on
this topic. First, for every $r$ odd, we construct a family of $r$-regular
graphs with palette index reaching the maximum admissible value. After that, we
construct the first known family of simple graphs whose palette index grows
quadratically with respect to their maximum degree.
