Bounds on Coloring Trees without Rainbow Paths
Wayne Goddard, Tyler Herrman, Simon J. Hughes

TL;DR
This paper studies the maximum number of colors in vertex colorings of trees that avoid rainbow paths of length 4 and 5, identifying extremal trees and calculating exact bounds.
Contribution
It extends the study of rainbow path parameters to trees for k ≥ 4, providing exact bounds and characterizing extremal trees such as coronas and octopuses.
Findings
Minimum c_4(T) and cp_4(T) is (n+2)/2, with coronas as extremal trees.
Minimum c_5(T) and cp_5(T) is (n+3)/2, with octopuses as extremal trees.
Exact values and extremal structures are determined for paths and general trees.
Abstract
For a graph with colored vertices, a rainbow subgraph is one where all vertices have different colors. For graph , let denote the maximum number of different colors in a coloring without a rainbow path on vertices, and the maximum number of colors if the coloring is required to be proper. The parameter has been studied by multiple authors. We investigate these parameters for trees and . We first calculate them when is a path, and determine when the optimal coloring is unique. Then for trees of order , we show that the minimum value of and is , and the trees with the minimum value of are the coronas. Further, the minimum value of and is , and the trees with the minimum value of either parameter are octopuses.
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems
