The $t$-Tone Chromatic Number of Classes of Sparse Graphs

TL;DR

This paper investigates the $t$-tone chromatic number of sparse graphs, providing exact values and bounds for various classes, including outerplanar graphs and cycles, advancing understanding of graph coloring constraints.

Contribution

It establishes sharp bounds and exact values for the $t$-tone chromatic number in specific classes of sparse graphs, including outerplanar graphs and cycles.

Findings

Exact $ au_2(G)$ for graphs with $ extrm{mad}(G)<12/5$

Bounds on $ au_2(G)$ for outerplanar graphs

Exact $ au_t(C_n)$ for $t=3,4,5$

Abstract

For a graph $G$ and $t,k\in\mathbb{Z}^+$ a \emph{$t$-tone $k$-coloring} of $G$ is a function $f:V(G)\rightarrow \binom{[k]}{t}$ such that $|f(v)\cap f(w)| < d(v,w)$ for all distinct $v,w \in V(G)$. The \emph{$t$-tone chromatic number} of $G$, denoted $\tau_t(G)$, is the minimum $k$ such that $G$ is $t$-tone $k$-colorable. For small values of $t$, we prove sharp or nearly sharp upper bounds on the $t$-tone chromatic number of various classes of sparse graphs. In particular, we determine $\tau_2(G)$ exactly when $\textrm{mad}(G) < 12/5$ and bound $\tau_2(G)$, up to a small additive constant, when $G$ is outerplanar. We also determine $\tau_t(C_n)$ exactly when $t\in\{3,4,5\}$.