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    Some results on the palette index of graphs

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