λ-backbone colorings along pairwise disjoint stars and matchings

Broersma, H.J.; Fujisawa, J.; Marchal, L.; Paulusma, D.; Salman, A.N.M.; Yoshimoto, K.

Abstract

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.

Citation

Broersma, H., Fujisawa, J., Marchal, L., Paulusma, D., Salman, A., & Yoshimoto, K. (2009). λ-backbone colorings along pairwise disjoint stars and matchings. Discrete Mathematics, 309(18), 5596-5609. https://doi.org/10.1016/j.disc.2008.04.007

• Journal Article Type: Article
• Publication Date: Sep 28, 2009
• Deposit Date: Mar 1, 2010
• Publicly Available Date: Jan 17, 2015
• Journal: Discrete mathematics.
• Print ISSN: 0012-365X
• Electronic ISSN: 2578-9252
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 309
• Issue: 18
• Pages: 5596-5609
• DOI: https://doi.org/10.1016/j.disc.2008.04.007
• Keywords: λ-backbone coloring, λ-backbone coloring number, Star, Matching.
• Public URL: https://durham-repository.worktribe.com/output/1545816