$t$-tone colorings of outerplanar and Halin graphs
Hadeel Al Bazzal, Olivier Togni

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 -tone -coloring of a graph assigns a set of distinct colors from to each vertex so that vertices at distance share fewer than common colors. The -tone chromatic number of is the minimum such that has a -tone -coloring. This paper investigates the -tone coloring of two specific subclasses of planar graphs: subcubic outerplanar graphs and Halin graphs. We provide a complete characterization of the -tone chromatic number for subcubic outerplanar graphs and establish a sharp upper bound for their -tone chromatic number. We then turn to Halin graphs and prove that every cubic Halin graph of order is -tone -colorable. Moreover, we derive an upper bound on the -tone chromatic number for Halin graphs with arbitrary maximum degree.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Graph Labeling and Dimension Problems
