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.

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 -