TY - JOUR

T1 - λ-backbone colorings along pairwise disjoint stars and matchings

AU - Broersma, H. J.

AU - Fujisawa, J.

AU - Marchal, L.

AU - Paulusma, D.

AU - Salman, A. N.M.

AU - Yoshimoto, K.

N1 - Funding Information:
We would like to thank the two anonymous referees for their useful feedback that helped in improving the presentation of our paper. The work of J.F. was partially supported by the 21st Century COE Program; Integrative Mathematical Sciences: Progress in Mathematics Motivated by Social and Natural Sciences.

PY - 2009/9/28

Y1 - 2009/9/28

N2 - 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.

AB - 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.

KW - Matching

KW - Star

KW - λ-backbone coloring

KW - λ-backbone coloring number

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U2 - 10.1016/j.disc.2008.04.007

DO - 10.1016/j.disc.2008.04.007

M3 - Article

AN - SCOPUS:69549108619

VL - 309

SP - 5596

EP - 5609

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 18

ER -