$t$-tone colorings of outerplanar and Halin graphs

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.

Figures (1)

• Figure 1: A subcubic outerplanar graph that is not $3$-tone $10$-colorable.

Theorems & Definitions (45)

• Definition 1.1
• Conjecture 1.1
• Theorem 1.1
• Lemma 2.1: BP
• Theorem 2.1
• proof
• Claim 2.1.1
• proof
• Definition 2.1
• Lemma 2.2