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λ-backbone colorings along pairwise disjoint stars and matchings.

Broersma, H. J. and Fujisawa, J. and Marchal, L. and Paulusma, D. and Salman, A. N. M. and Yoshimoto, K. (2009) 'λ-backbone colorings along pairwise disjoint stars and matchings.', Discrete mathematics., 309 (18). pp. 5596-5609.


Given an integer λ≥2, a graph G=(V,E) and a spanning subgraph H of G (the backbone of G), a λ-backbone coloring of (G,H) is a proper vertex coloring V→{1,2,…} of G, in which the colors assigned to adjacent vertices in H differ by at least λ. We study the case where the backbone is either a collection of pairwise disjoint stars or a matching. We show that for a star backbone S of G the minimum number ℓ for which a λ-backbone coloring of (G,S) with colors in {1,…,ℓ} exists can roughly differ by a multiplicative factor of at most View the MathML source from the chromatic number χ(G). For the special case of matching backbones this factor is roughly View the MathML source. We also show that the computational complexity of the problem “Given a graph G with a star backbone S, and an integer ℓ, is there a λ-backbone coloring of (G,S) with colors in {1,…,ℓ}?” jumps from polynomially solvable to NP-complete between ℓ=λ+1 and ℓ=λ+2 (the case ℓ=λ+2 is even NP-complete for matchings). We finish the paper by discussing some open problems regarding planar graphs.

Item Type:Article
Keywords:λ-backbone coloring, λ-backbone coloring number, Star, Matching.
Full text:(AM) Accepted Manuscript
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Publisher statement:NOTICE: this is the author’s version of a work that was accepted for publication in Discrete mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Discrete mathematics, 309/18, 2009, 10.1016/j.disc.2008.04.007
Date accepted:No date available
Date deposited:17 January 2015
Date of first online publication:September 2009
Date first made open access:No date available

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