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Some progress on $t$-tone coloring

Patrick Bennett, Jade Nichols

TL;DR

This work advances the theory of t-tone coloring by giving tight asymptotic bounds for large families of graphs and large-t regimes. It develops a unified hypergraph-matching framework to derive coloring bounds for trees, sparse random graphs, complete multipartite graphs, and Cartesian powers, and it obtains precise large-t formulas τ_t(G)=tn−κ_G with equality under diameter-dependent thresholds. The results include near-optimal bounds for trees with large maximum degree, asymptotics for random graphs G_{n,p}, and tight behavior for Cartesian powers and K_n^b, plus connections to MOLS for exact K_n^2 colorings. The paper also proposes conjectures about tree coloring and demonstrates evidence through explicit small-case computations, suggesting rich structural patterns and inviting further hypergraph-methods-based exploration.

Abstract

A $t$-tone coloring of a graph $G$ assigns to each vertex a set of $t$ colors such that any pair of vertices $u, v$ with distance $d$ can share at most $d-1$ colors. In this note, we prove several new results on $t$-tone coloring. For example we prove a new result for trees of large maximum degree, as well as some results for the cartesian power of a graph. We also make a conjecture about trees.

Some progress on $t$-tone coloring

TL;DR

This work advances the theory of t-tone coloring by giving tight asymptotic bounds for large families of graphs and large-t regimes. It develops a unified hypergraph-matching framework to derive coloring bounds for trees, sparse random graphs, complete multipartite graphs, and Cartesian powers, and it obtains precise large-t formulas τ_t(G)=tn−κ_G with equality under diameter-dependent thresholds. The results include near-optimal bounds for trees with large maximum degree, asymptotics for random graphs G_{n,p}, and tight behavior for Cartesian powers and K_n^b, plus connections to MOLS for exact K_n^2 colorings. The paper also proposes conjectures about tree coloring and demonstrates evidence through explicit small-case computations, suggesting rich structural patterns and inviting further hypergraph-methods-based exploration.

Abstract

A -tone coloring of a graph assigns to each vertex a set of colors such that any pair of vertices with distance can share at most colors. In this note, we prove several new results on -tone coloring. For example we prove a new result for trees of large maximum degree, as well as some results for the cartesian power of a graph. We also make a conjecture about trees.
Paper Structure (15 sections, 22 theorems, 45 equations, 1 figure)

This paper contains 15 sections, 22 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Trees
  3. Sparse random graphs
  4. Complete multipartite graphs
  5. 2-tone coloring graphs with small chromatic number
  6. Cartesian powers
  7. $t$-tone coloring a fixed graph $G$ for large $t$
  8. Organization of paper
  9. Terminology and tools
  10. Trees, sparse random graphs, and complete multipartite graphs
  11. 2-tone coloring graphs with small chromatic number
  12. Cartesian powers
  13. $t$-tone coloring a fixed graph $G$ for large $t$
  14. Evidence supporting Conjecture \ref{['conj:tree']}
  15. Concluding remarks

Key Result

Theorem 1

If $T$ is a tree with $\Delta(T)=\Delta$ then

Figures (1)

  • Figure 1: How to 3-tone color $T_7$

Theorems & Definitions (39)

  • Theorem 1: FGPS
  • proof
  • Theorem 2: CKK13
  • Theorem 3
  • Conjecture 1
  • Theorem 4: BBDF
  • Theorem 5
  • Corollary 1
  • Theorem 6: LMM
  • Theorem 7
  • ...and 29 more