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