Some progress on $t$-tone coloring

Patrick Bennett; Jade Nichols

Some progress on $t$-tone coloring

Combinatorics 2025-10-16 v1 Discrete Mathematics

Authors: Patrick Bennett , Jade Nichols

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.

Keywords

graph coloring ramsey theory and colorings rainbow coloring

Cite

@article{arxiv.2510.13382,
  title  = {Some progress on $t$-tone coloring},
  author = {Patrick Bennett and Jade Nichols},
  journal= {arXiv preprint arXiv:2510.13382},
  year   = {2025}
}

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