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 $\frac{n}{2}$.

翻译：图 $G$ 及其子图（亦称主干）$H$ 的 $\lambda$-主干着色是指函数 $c \colon V(G) \rightarrow \{1,\dots, k\}$，满足 $c$ 是 $G$ 的正常着色，且对于每条边 $\{u,v\} \in E(H)$，有 $|c(u) - c(v)| \ge \lambda$。本文提出了一种在线性时间内对以树和森林为背板的团图进行着色的方法，使得最大颜色数不超过 $\max\{n, 2 \lambda\} + \Delta(H)^2 \lceil\log{n} \rceil$。该结果改进了已有的近似算法，因为它实现了 $(\Delta(H)^2 \lceil\log{n} \rceil)$-绝对近似，即在最优解基础上具有加法误差。我们还给出了一个无限树族 $T$，其 $\Delta(T) = 3$，对于接近 $\frac{n}{2}$ 的 $\lambda$，以 $T$ 为背板的团图着色至少需要 $\max\{n, 2 \lambda\} + \Omega(\log{n})$ 种颜色。