Classify trees minimizing the proper no-rainbow-P4 coloring parameter

Determine all trees that attain the minimum possible value of the maximum number of colors in a proper vertex-coloring with no rainbow path on four vertices.

Background

The paper studies the maximum number of colors permitted in vertex-colorings of graphs that contain no rainbow path on a prescribed number of vertices, both with and without the requirement that the coloring be proper. For trees and four-vertex paths, it proves the minimum value of the relevant proper parameter but does not classify every tree attaining that minimum.

The paper establishes that coronas are precisely the trees minimizing the corresponding nonproper parameter, while noting that the proper-parameter extremal trees do not appear to have a similarly simple description. The conclusion therefore identifies the complete classification of trees attaining the minimum proper value as an unresolved problem.