On λ-backbone coloring of cliques with tree backbones in linear time

TL;DR

The paper studies BBC_\lambda(K_n, F) for complete graphs with tree or forest backbones and presents a linear-time algorithm that achieves $BBC_{\lambda}(K_n, F) \le \max\{n, 2 \lambda\} + \Delta(F)^2 \lceil \log n \rceil$, offering a strong additive-approximation guarantee. It introduces a red-blue-yellow decomposition framework to structure color assignments and proves the bound for trees and forests, with linear-time construction. To assess tightness, it constructs an infinite family of Δ=3 Fibonacci-based trees showing a matching lower bound of $\Theta(\log n)$ over the base $\max\{n, 2\lambda\}$, illustrating that the additive term can be necessary. The work also connects backbone coloring to split graphs and highlights open questions on improving the additive term and extending the approach to broader graph classes.

Abstract

A $λ$-backbone coloring of a graph $G$ with its subgraph (also called a backbone) $H$ is a function $c \colon V(G) \rightarrow \{1,\dots, k\}$ ensuring that $c$ is a proper coloring of $G$ and for each $\{u,v\} \in E(H)$ it holds that $|c(u) - c(v)| \ge λ$. In this paper we propose a way to color cliques with tree and forest backbones in linear time that the largest color does not exceed $\max\{n, 2 λ\} + Δ(H)^2 \lceil\log{n} \rceil$. This result improves on the previously existing approximation algorithms as it is $(Δ(H)^2 \lceil\log{n} \rceil)$-absolutely approximate, i.e. with an additive error over the optimum. We also present an infinite family of trees $T$ with $Δ(T) = 3$ for which the coloring of cliques with backbones $T$ require to use at least $\max\{n, 2 λ\} + Ω(\log{n})$ colors for $λ$ close to $\frac{n}{2}$.

Figures (3)

• Figure 1: The main idea for the $\lambda$-backbone coloring based on a red-blue-yellow decomposition. For brevity, $M = 2 L + D^2 |Y_1| + D |Y_2|$ denotes the maximum color used.
• Figure 2: Fibonacci trees for $N = 1, 2, 3, 4, 5$. White vertices are the roots of Fibonacci trees and gray vertices are the roots of their Fibonacci subtrees.
• Figure 3: An example Fibonacci tree $T^F_n$ with $|Y_1| = 2$ and the imbalance of the top tree $imb(T_1) = |Y_1| + F_n - F_{n - 1} - F_{n - 2} = 2$.

Theorems & Definitions (45)

• definition 1
• definition 2
• theorem 1: Broersma et al., broersma2007backbone
• theorem 2: Havet et al., havet2014circular
• theorem 3: Janczewski, Turowski, janczewski2015backbone
• definition 3
• lemma 1
• proof
• definition 4
• lemma 2