Coloring the square of a planar graph

Heuvel, Jan van den; McGuinness, Sean

doi:10.1002/jgt.10077

Cited by 142 publications

(66 citation statements)

References 11 publications

(4 reference statements)

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“…Much of the research has been concentrated on the case that G 1 is a planar graph. We refer to [1,3,4,18,21,22] for more details. In some versions of this problem one puts the additional restriction on G 1 that the colors should be sufficiently separated, in order to model practical frequency assignment problems in which interference should be kept at an acceptable level.…”

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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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: 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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“…Much of the research has been concentrated on the case that G 1 is a planar graph. We refer to [1], [3], [4], [16], [19], and [20] for more details. In some versions of this problem one puts the additional restriction on G 1 that the colors should be sufficiently separated, in order to model practical frequency assignment problems in which interference should be kept at an acceptable level.…”

Section: Introduction and Related Researchmentioning

confidence: 99%

Backbone Colorings for Networks

Broersma

Fomin

Golovach

et al. 2003

Graph-Theoretic Concepts in Computer Science

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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 . For the special case that G is a split graph the difference from χ(G) is at most an additive constant 2 or 1, for tree backbones and path backbones, respectively. 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).

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“…For another basic case of p = q (i.e., it is equivalent to L(1, 1)-labeling problem), it is also shown to be NP-hard in [54], where the problem is called Distance-2 Graph Coloring Problem. The L(1, 1)-labeling problem is well studied also from the viewpoint of restricted classes of graphs, such as planar graphs [2,10,55,60,66], outerplanar graphs [3], and so on [8,53].…”

Section: The Computational Complexity Of L(p Q)-labeling Problemsmentioning

confidence: 99%

Algorithmic aspects of distance constrained labeling: a survey

Hasunuma

Ishii

Ono

et al. 2014

IJNC

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Distance constrained labeling problems, e.g., L(p, q)-labeling and (p, q)-total labeling, are originally motivated by the frequency assignment. From the viewpoint of theory, the upper bounds on the labeling numbers and the time complexity of finding a minimum labeling are intensively and extensively studied. In this paper, we survey the distance constrained labeling problems from algorithmic aspects, that is, computational complexity, approximability, exact computation, and so on.

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Coloring the square of a planar graph

Abstract: In 1977, Wegner conjectured that the chromatic number of the square of every planar graph G with maximum degree ∆ ≥ 8 is at most 3 2 ∆ + 1. We show that it is at most 3 2 ∆ (1 + o(1)).

Cited by 142 publications

References 11 publications

Backbone colorings for graphs: Tree and path backbones

Backbone colorings for graphs: Tree and path backbones

Backbone Colorings for Networks

Algorithmic aspects of distance constrained labeling: a survey

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