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

A $\lambda$-backbone coloring of a graph $G$ with its subgraph (also called a backbone) $H$ is a function $c:V\left(G\right)\to \left\{1,\dots ,k\right\}$ ensuring that $c$ is a proper coloring of $G$ and for each $\left\{u,v\right\}\in E\left(H\right)$ it holds that $|c\left(u\right)-c\left(v\right)|\ge \lambda$. 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 $\max \left\{n,2\lambda \right\}+\Delta {\left(H\right)}^2\lceil \log n\rceil$. This result improves on the previously existing approximation algorithms as it is $\left(\Delta {\left(H\right)}^2\lceil \log n\rceil \right)$-absolutely approximate, that is, with an additive error over the optimum. We also present an infinite family of trees $T$ with $\Delta \left(T\right)=3$ for which the coloring of cliques with backbones $T$ requires at least $\max \left\{n,2\lambda \right\}+\Omega \left(\log n\right)$ colors for $\lambda$ close to $\frac{n}{2}$. The construction draws on the theory of Fibonacci numbers, particularly on Zeckendorf representations.