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

Abstract

AbstractA ‐backbone coloring of a graph with its subgraph (also called a backbone) is a function ensuring that is a proper coloring of and for each it holds that . 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 . This result improves on the previously existing approximation algorithms as it is ‐absolutely approximate, that is, with an additive error over the optimum. We also present an infinite family of trees with for which the coloring of cliques with backbones requires at least colors for close to . The construction draws on the theory of Fibonacci numbers, particularly on Zeckendorf representations.