The t-tone chromatic number of random graphs

TL;DR

This paper investigates the t-tone chromatic number of random graphs, establishing asymptotic relationships with the chromatic number for various regimes of edge probability, and extends results to general t-tone colorings.

Contribution

It provides the first probabilistic bounds on the t-tone chromatic number of random graphs, linking it to the chromatic number and maximum degree in different regimes.

Findings

For dense graphs, τ₂(G) ≈ 2χ(G) with high probability.

For sparse graphs, τ₂(G) depends on the maximum degree Δ.

Results extend to general t-tone colorings.

Abstract

A proper 2-tone $k$-coloring of a graph is a labeling of the vertices with elements from $\binom{[k]}{2}$ such that adjacent vertices receive disjoint labels and vertices distance 2 apart receive distinct labels. The 2-tone chromatic number of a graph $G$, denoted $\tau_2(G)$ is the smallest $k$ such that $G$ admits a proper 2-tone $k$ coloring. In this paper, we prove that w.h.p. for $p\ge Cn^{-1/4}\ln^{9/4}n$, $\tau_2(G_{n,p})=(2+o(1))\chi(G_{n,p})$ where $\chi$ represents the ordinary chromatic number. For sparse random graphs with $p=c/n$, $c$ constant, we prove that $\tau_2(G_{n,p}) = \lceil{{\sqrt{8\Delta+1} +5}/{2}}\rceil$ where $\Delta$ represents the maximum degree. For the more general concept of $t$-tone coloring, we achieve similar results.