On λ-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 $c \colon V(G) \rightarrow {1,\dots, k}$ ensuring that $c$ is a proper coloring of $G$ and for each ${u,v} \in E(H)$ it holds that $|c(u) - c(v)| \ge \lambda$. 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 $\max{n, 2 \lambda} + \Delta(H)² \lceil\log{n} \rceil$. This result improves on the previously existing approximation algorithms as it is $(\Delta(H)² \lceil\log{n} \rceil)$-absolutely approximate, i.e. with an additive error over the optimum. We also present an infinite family of trees $T$ with $\Delta(T) = 3$ for which the coloring of cliques with backbones $T$ require to use at least $\max{n, 2 \lambda} + \Omega(\log{n})$ colors for $\lambda$ close to $\frac{n}{2}$.