Some results on the palette index of graphs

Given a proper edge coloring φ of a graph G, we define the palette SG(υ, φ) of a vertex υ ∈ V (G) as the set of all colors appearing on edges incident with υ. The palette index š(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 š(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.