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Bounds on Coloring Trees without Rainbow Paths

Wayne Goddard, Tyler Herrman, Simon J. Hughes

TL;DR

This work studies coloring graphs to avoid rainbow copies of a fixed path P_k, focusing on trees and k ≥ 4. It derives exact formulas for colorings on paths: for any n ≥ 1, c_k(P_n) = ⌊(k-2)n/(k-1)⌋ + 1 and cp_k(P_n) = ⌊((k-3)n+1)/(k-2)⌋ + 1, with precise uniqueness conditions, using P_m-attaching lemmas and thwarting-set arguments. For trees, it establishes that the minimum values for c_4(T) and cp_4(T) are (n+2)/2, with coronas uniquely attaining cp_4-minimization, while c_4-minimization is achieved by coronas as well, and shows the minimums for c_5(T) and cp_5(T) are (n+3)/2, achieved by octopuses; octopuses are also characterized as the unique extremal trees for odd n in the k = 5 case. The results combine structural arguments (via thwarting sets and boring-colorings) with explicit tree constructions (coronas and octopuses) to provide tight, combinatorial bounds and highlight distinct extremal behaviors for k = 4 and k = 5, suggesting directions for broader graph classes and larger k.

Abstract

For a graph with colored vertices, a rainbow subgraph is one where all vertices have different colors. For graph $G$, let $c_k(G)$ denote the maximum number of different colors in a coloring without a rainbow path on $k$ vertices, and $cp_k(G)$ the maximum number of colors if the coloring is required to be proper. The parameter $c_3$ has been studied by multiple authors. We investigate these parameters for trees and $k \ge 4$. We first calculate them when $G$ is a path, and determine when the optimal coloring is unique. Then for trees $T$ of order $n$, we show that the minimum value of $c_4(T)$ and $cp_4(T)$ is $(n+2)/2$, and the trees with the minimum value of $cp_4(T)$ are the coronas. Further, the minimum value of $c_5(T)$ and $cp_5(T)$ is $(n+3)/2$ , and the trees with the minimum value of either parameter are octopuses.

Bounds on Coloring Trees without Rainbow Paths

TL;DR

This work studies coloring graphs to avoid rainbow copies of a fixed path P_k, focusing on trees and k ≥ 4. It derives exact formulas for colorings on paths: for any n ≥ 1, c_k(P_n) = ⌊(k-2)n/(k-1)⌋ + 1 and cp_k(P_n) = ⌊((k-3)n+1)/(k-2)⌋ + 1, with precise uniqueness conditions, using P_m-attaching lemmas and thwarting-set arguments. For trees, it establishes that the minimum values for c_4(T) and cp_4(T) are (n+2)/2, with coronas uniquely attaining cp_4-minimization, while c_4-minimization is achieved by coronas as well, and shows the minimums for c_5(T) and cp_5(T) are (n+3)/2, achieved by octopuses; octopuses are also characterized as the unique extremal trees for odd n in the k = 5 case. The results combine structural arguments (via thwarting sets and boring-colorings) with explicit tree constructions (coronas and octopuses) to provide tight, combinatorial bounds and highlight distinct extremal behaviors for k = 4 and k = 5, suggesting directions for broader graph classes and larger k.

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.
Paper Structure (7 sections, 17 theorems, 1 equation, 6 figures)

This paper contains 7 sections, 17 theorems, 1 equation, 6 figures.

Table of Contents

  1. Introduction
  2. Colorings Paths Without Rainbow Paths
  3. Coloring Trees Without a Rainbow $P_4$
  4. Trees with Extremal ${\mathit{c}}_4$
  5. Trees with Minimum ${\mathit{cp}}_4$
  6. Coloring Trees Without a Rainbow $P_5$
  7. Conclusion

Key Result

Lemma 1

(a) Assume $k\ge 2$. For any graph $G$ and vertex $w$, if graph $G_1$ is obtained from $G$ by attaching $P_{k-1}$ to $w$, then ${\mathit{c}}_{k} ( G_1 ) = {\mathit{c}}_k (G ) + k- 2$. (b) Assume $k\ge 3$. For any graph $G$ and end-vertex$w$, if graph $G_2$ is obtained from $G$ by attaching $P_{k-2}

Figures (6)

  • Figure 1: Attaching a $P_3$
  • Figure 2: A corona and a minimum $P_4$-thwarting set
  • Figure 3: The double star $D_2$ with ${\mathit{cp}}_4 = 4$
  • Figure 4: A possible coloring
  • Figure 5: A tree with ${\mathit{cp}}_4(T) = 6$
  • ...and 1 more figures

Theorems & Definitions (17)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 2
  • Theorem 5
  • Lemma 3
  • Theorem 6
  • Theorem 7
  • ...and 7 more