Approximations for  -Colorings of Graphs

Bodlaender, Hans L.; Kloks, Ton; Tan, Richard B.; Leeuwen, Jan van

doi:10.1093/comjnl/47.2.193

Cited by 99 publications

(103 citation statements)

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“…Similarly as in (1), the values P(k) := max {bbc(G, P) : G a graph with Hamiltonian path P, and χ(G) = k} (2) are considered. In Section 3, we will exactly determine all these values P(k) and observe that they roughly grow like 3k/2.…”

Section: B Resultsmentioning

confidence: 99%

“…One way to model this is to use positive integers for the colors (modeling certain frequency channels) and to ask for a coloring of G 1 and G 2 such that the colors on adjacent vertices in G 2 are different, whereas they differ by at least 2 on adjacent vertices in G 1 . This problem is known as the L(2, 1)-labeling problem and has been studied (under various names) in [2,[7][8][9][10][11]19].…”

Section: Backbone Colorings For Graphs 139mentioning

confidence: 99%

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Backbone colorings for graphs: Tree and path backbones

Broersma

Fomin

Golovach

et al. 2007

Journal of Graph Theory

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Abstract:We introduce and study backbone colorings, a variation on classical vertex colorings: Given a graph G = (V, E) and a spanning subgraph H of G (the backbone of G), a backbone coloring for G and 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 two. We study the cases where the backbone is either a spanning tree or a spanning path. We show that for tree backbones of G the number of colors needed for a backbone coloring of G can roughly differ by a multiplicative factor of at most 2 from the chromatic number χ(G); for path backbones this factor is roughly 3 2. We show that the computational complexity of the problem "Given a graph G, a spanning tree T of G, and an integer , is there a backbone coloring for G and T with at most colors?" jumps from polynomial to NP-complete between = 4 (easy for all spanning trees) and = 5 (difficult even for spanning paths). We finish the paper by discussing some open problems.

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Section: B Resultsmentioning

confidence: 99%

Section: Backbone Colorings For Graphs 139mentioning

confidence: 99%

Backbone colorings for graphs: Tree and path backbones

Broersma

Fomin

Golovach

et al. 2007

Journal of Graph Theory

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“…The L(1, 1)-labeling problem appeared in [3,6] and the L(2, 1)-labeling in [3,6,9]. To the best of our knowledge, nothing is known for the L(h, 1, 1)-labeling of outerplanar graphs for h ≥ 2.…”

Section: Related Resultsmentioning

confidence: 99%

L(h, 1, 1)-labeling of outerplanar graphs

et al. 2008

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Abstract. An L(h, 1, 1)-labeling of a graph is an assignment of labels from the set of integers {0, · · · , λ} to the vertices of the graph such that adjacent vertices are assigned integers of at least distance h ≥ 1 apart and all vertices of distance three or less must be assigned different labels. The aim of the L(h, 1, 1)-labeling problem is to minimize λ, denoted by λ h,1,1 and called span of the L(h, 1, 1)-labeling. As outerplanar graphs have bounded treewidth, the L(1, 1, 1)-labeling problem on outerplanar graphs can be exactly solved in O(n 3 ), but the multiplicative factor depends on the maximum degree ∆ and is too big to be of practical use. In this paper we give a linear time approximation algorithm for computing the more general L (h, 1, 1)-labeling for outerplanar graphs that is within additive constants of the optimum values.

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“…One way to model this is to use positive integers for the colors (modeling certain frequency channels) and to ask for a coloring of G 1 and G 2 such that the colors on adjacent vertices in G 2 are different, whereas they differ by at least 2 on adjacent vertices in G 1 . A closely related variant is known as the radio coloring problem and has been studied (under various names) in [2], [10], [11], [12], [13], [14], and [21].…”

Section: Introduction and Related Researchmentioning

confidence: 99%

Backbone colorings along stars and matchings in split graphs: their span is close to the chromatic number

Broersma

Marchal²,

Paulusma

et al. 2009

Discuss. Math. Graph Theory

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We continue the study on backbone colorings, a variation on classical vertex colorings that was introduced at WG2003. Given a graph G = (V, E) and a spanning subgraph H of G (the backbone of G), a λ-backbone coloring for G and 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. The algorithmic and combinatorial properties of backbone colorings have been studied for various types of backbones in a number of papers. The main outcome of earlier studies is that the minimum number ℓ of colors for which such colorings V → {1, 2, . . . , ℓ} exist in the worst case is a factor times the chromatic number (for all studied types of backbones). We show here that for split graphs and matching or star backbones, ℓ is at most a small additive constant (depending on λ) higher than the chromatic number. Our proofs combine algorithmic and combinatorial arguments. We also indicate other graph classes for which our results imply better upper bounds on ℓ than the previously known bounds.

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Approximations for -Colorings of Graphs

Cited by 99 publications

References 0 publications

Backbone colorings for graphs: Tree and path backbones

Backbone colorings for graphs: Tree and path backbones

L(h, 1, 1)-labeling of outerplanar graphs

Backbone colorings along stars and matchings in split graphs: their span is close to the chromatic number

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