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

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 $\frac{n}{2}$.