On λ-backbone coloring of cliques with tree backbones in linear time
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A λ-backbone coloring of a graph G with its subgraph (also called a backbone) H is a function c V(G) →{1,…, k} ensuring that c is a proper coloring of G and for each {u,v}∈ E(H) it holds that |c(u) - c(v)| ≥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 ⌈logn⌉. This result improves on the previously existing approximation algorithms as it is (Δ(H)^2 ⌈logn⌉)-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(logn) colors for λ close to n/2.
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