Given a proper edge coloring φ of a graph G, we define the palette
SG(v,φ) of a vertex v∈V(G) as the set of all colors appearing
on edges incident with v. The palette index sˇ(G) of G is the
minimum number of distinct palettes occurring in a proper edge coloring of G.
In this paper we give various upper and lower bounds on the palette index of
G in terms of the vertex degrees of G, particularly for the case when G
is a bipartite graph with small vertex degrees. Some of our results concern
(a,b)-biregular graphs; that is, bipartite graphs where all vertices in one
part have degree a and all vertices in the other part have degree b. We
conjecture that if G is (a,b)-biregular, then sˇ(G)≤1+max{a,b}, and we prove that this conjecture holds for several families of
(a,b)-biregular graphs. Additionally, we characterize the graphs whose
palette index equals the number of vertices