λ-backbone colorings along pairwise disjoint stars and matchings

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

Research output: Contribution to journalArticle

17 Citations (Scopus)

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 2 - frac(1, λ) from the chromatic number χ (G). For the special case of matching backbones this factor is roughly 2 - frac(2, λ + 1). 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.

Original languageEnglish
Pages (from-to)5596-5609
Number of pages14
JournalDiscrete Mathematics
Volume309
Issue number18
DOIs
Publication statusPublished - 2009 Sep 28

Keywords

  • Matching
  • Star
  • λ-backbone coloring
  • λ-backbone coloring number

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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    Broersma, H. J., Fujisawa, J., Marchal, L., Paulusma, D., Salman, A. N. M., & 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