$t$-tone colorings of outerplanar and Halin graphs

TL;DR

This paper studies $t$-tone colorings of specific planar graph subclasses, providing exact and upper bound results for outerplanar and Halin graphs, advancing understanding of their coloring properties.

Contribution

It characterizes the 2-tone chromatic number for subcubic outerplanar graphs and establishes bounds for their 3-tone chromatic number, also analyzing Halin graphs' 2-tone colorability.

Findings

Complete characterization of 2-tone chromatic number for subcubic outerplanar graphs.

Sharp upper bound for 3-tone chromatic number of subcubic outerplanar graphs.

Every cubic Halin graph with at least 6 vertices is 2-tone 7-colorable.

Abstract

A $t$-tone $k$-coloring of a graph $G$ assigns a set of $t$ distinct colors from $\{1, \dots, k\}$ to each vertex so that vertices at distance $d$ share fewer than $d$ common colors. The $t$-tone chromatic number of $G$ is the minimum $k$ such that $G$ has a $t$-tone $k$-coloring. This paper investigates the $t$-tone coloring of two specific subclasses of planar graphs: subcubic outerplanar graphs and Halin graphs. We provide a complete characterization of the $2$-tone chromatic number for subcubic outerplanar graphs and establish a sharp upper bound for their $3$-tone chromatic number. We then turn to Halin graphs and prove that every cubic Halin graph of order $n \ge 6$ is $2$-tone $7$-colorable. Moreover, we derive an upper bound on the $2$-tone chromatic number for Halin graphs with arbitrary maximum degree.