Some results on the palette index of graphs
C.J. Casselgren, Petros A. Petrosyan

TL;DR
This paper investigates the palette index of graphs, providing bounds and conjectures, especially for bipartite graphs with small degrees, and characterizes graphs with maximum palette index.
Contribution
It introduces new bounds on the palette index based on vertex degrees and proves the conjecture for certain bipartite graph families.
Findings
Bounds on palette index in terms of vertex degrees
Conjecture that palette index for bipartite graphs is at most 1 plus the maximum degree
Characterization of graphs with palette index equal to the number of vertices
Abstract
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…
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