$t$-tone edge coloring of graphs
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914596407738368 |
|---|---|
| author | Bazzal, Hadeel Al |
| author_facet | Bazzal, Hadeel Al |
| contents | In this paper, we introduce the notion of $t$-tone edge coloring. A $t$-tone edge $k$-coloring of a graph $G$ assigns to each edge of $G$ a set of $t$ distinct colors from $\{1,\dots,k\}$ such that any two edges at distance $d$ share fewer than $d$ common colors. The $t$-tone chromatic index of $G$, denoted by $τ'_t(G)$, is the minimum integer $k$ for which $G$ admits a $t$-tone edge $k$-coloring. We focus on the case $t=2$ and establish several upper bounds on $τ'_2$. In particular, for every graph $G$ with maximum degree $Δ(G)\ge2$, we prove that $τ'_2(G)\le 6Δ(G)-4$, improving the corresponding bound derived from the vertex analogue. We also show that every tree $T$ with $Δ(T)\ge3$ satisfies $τ'_2(T)=2Δ(T)$. Furthermore, every planar graph $G$ satisfies $τ'_2(G)\le \max\{41,3Δ(G)+5\}$, while every outerplanar graph $G$ satisfies $τ'_2(G)\le \max\{14,3Δ(G)\}$. For subcubic graphs $G$, the vertex analogue yields $τ'_2(G)\le12$. We improve this bound to $11$ for claw-free subcubic graphs and to $10$ for $2$-degenerate subcubic graphs. Finally, we propose two conjectures concerning optimal bounds for cubic and $K_4$-free cubic graphs, and establish them for series-parallel subcubic multigraphs and subcubic outerplanar graphs, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24571 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $t$-tone edge coloring of graphs Bazzal, Hadeel Al Combinatorics In this paper, we introduce the notion of $t$-tone edge coloring. A $t$-tone edge $k$-coloring of a graph $G$ assigns to each edge of $G$ a set of $t$ distinct colors from $\{1,\dots,k\}$ such that any two edges at distance $d$ share fewer than $d$ common colors. The $t$-tone chromatic index of $G$, denoted by $τ'_t(G)$, is the minimum integer $k$ for which $G$ admits a $t$-tone edge $k$-coloring. We focus on the case $t=2$ and establish several upper bounds on $τ'_2$. In particular, for every graph $G$ with maximum degree $Δ(G)\ge2$, we prove that $τ'_2(G)\le 6Δ(G)-4$, improving the corresponding bound derived from the vertex analogue. We also show that every tree $T$ with $Δ(T)\ge3$ satisfies $τ'_2(T)=2Δ(T)$. Furthermore, every planar graph $G$ satisfies $τ'_2(G)\le \max\{41,3Δ(G)+5\}$, while every outerplanar graph $G$ satisfies $τ'_2(G)\le \max\{14,3Δ(G)\}$. For subcubic graphs $G$, the vertex analogue yields $τ'_2(G)\le12$. We improve this bound to $11$ for claw-free subcubic graphs and to $10$ for $2$-degenerate subcubic graphs. Finally, we propose two conjectures concerning optimal bounds for cubic and $K_4$-free cubic graphs, and establish them for series-parallel subcubic multigraphs and subcubic outerplanar graphs, respectively. |
| title | $t$-tone edge coloring of graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.24571 |