$t$-tone colorings of outerplanar and Halin graphs

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.

💡 Research Summary

The paper studies t‑tone colorings, a generalization of ordinary vertex colorings in which each vertex receives a set of t distinct colors from {1,…,k} and any two vertices at distance d share fewer than d colors. The smallest k for which such a coloring exists is the t‑tone chromatic number τ_t(G). While τ_t has been investigated for several graph families, the authors focus on two planar subclasses: subcubic outerplanar graphs (Δ≤3 and all vertices lie on the outer face) and Halin graphs (a tree with no degree‑2 vertices together with a cycle joining its leaves).

Subcubic outerplanar graphs.
The authors introduce the notion of a “good” 2‑tone k‑coloring, where vertices at distance two share exactly one color, and a “3‑good” k‑coloring, where adjacent vertices share no colors and distance‑two vertices share exactly one. Using these strengthened models they obtain a complete classification of τ_2 for connected subcubic outerplanar graphs of order n≥3:

• τ_2(G)=5 if G contains none of the cycles C₃, C₄, C₇.
• τ_2(G)=6 if G contains at least one of C₃, C₄, C₇ but does not contain K₄−e (the complete graph K₄ with one edge removed).
• τ_2(G)=7 otherwise (i.e., G contains K₄−e).

The proof proceeds by minimal counterexample arguments, exploiting the fact that the weak dual of an outerplanar graph is a forest. Pendant faces are removed, the remaining graph is colored inductively, and the removed vertices are re‑inserted with carefully chosen color sets. The classification settles the τ_2 problem for this class and confirms part (3) of the Bickle‑Phillips conjecture for subcubic outerplanar graphs.

For τ_3, rather than handling the full distance‑three constraints directly, the authors work with 3‑good colorings. They prove that every subcubic outerplanar graph admits a 3‑good 11‑coloring, which immediately yields τ_3(G)≤11. This bound improves upon previously known general bounds for τ_3 on planar graphs and is shown to be sharp for certain constructions.

Halin graphs.
A Halin graph H = T ∪ C consists of a tree T (with no degree‑2 vertices) embedded in the plane and a cycle C that connects the leaves of T in their cyclic order. The paper first treats cubic Halin graphs (all internal vertices have degree three). By constructing a 2‑tone coloring of the tree part with five colors and extending it around the leaf cycle, the authors prove that every cubic Halin graph with n≥6 satisfies τ_2(H)≤7. This result is tight because K₄−e forces τ_2≥7.

For Halin graphs with arbitrary maximum degree Δ, the authors derive an upper bound of order √Δ. More precisely, they show τ_2(H) ≤ ⌈√(2Δ)⌉ + c for a small constant c (explicitly given in the paper). This improves the generic planar bound τ_2 ≤ O(Δ) and matches the asymptotic behavior known for certain dense planar families. The proof uses a coloring of the underlying tree with Δ colors and a careful recoloring of the leaf cycle to respect the distance‑two intersection condition.

Further conjecture.
Motivated by the success of the “good” colorings, the authors conjecture a general relationship between successive t‑tone chromatic numbers for subcubic outerplanar graphs: τ_t(G) ≤ τ_{t‑1}(G) + c_t, where c_t depends only on t. This suggests a linear growth of the required palette as t increases, a direction for future research.

Significance.
The paper delivers the first exact determination of τ_2 for subcubic outerplanar graphs and a sharp universal bound for τ_3 on the same class. For Halin graphs, it provides the best known τ_2 bounds, especially the √Δ asymptotic for arbitrary degree. The introduction of “good” colorings offers a powerful technique for handling distance‑based constraints, potentially applicable to other graph families. Overall, the work advances the understanding of t‑tone colorings in planar settings and opens several avenues for further investigation.