Polychromatic colorings of bounded degree plane graphs

Aigner-Horev, Elad; Krakovski, Roi

doi:10.1002/jgt.20357

Cited by 15 publications

(11 citation statements)

References 6 publications

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“…Our main result bounds the minimum possible polychromatic number for plane graphs G with g(G) = g. This settles a question raised in [9]. Note that the set 3g− 5 4 , .…”

Section: Introductionsupporting

confidence: 66%

“…Similarly, a simple consequence of Theorem 2 is the following: Hoffmann and Kriegel [8] proved that the polychromatic number of any plane, bipartite, 2-connected simple graph is at least 3. Horev and Krakovski [9] showed that any connected plane multigraph G with g(G) ≥ 3 and maximum degree at most 3, which is not K 4 , can be colored with 3 colors such that every bounded face of G is polychromatic.…”

Section: Related Workmentioning

confidence: 99%

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Polychromatic colorings of plane graphs

Alon

Berke

Buchin

et al. 2008

Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry

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We show that the vertices of any plane graph in which every face is of length at least g can be colored by ⌊(3g − 5)/4⌋ colors so that every color appears in every face. This is nearly tight, as there are plane graphs that admit no vertex coloring of this type with more than ⌊(3g + 1)/4⌋ colors. We further show that the problem of determining whether a plane graph admits a vertex coloring by 3 colors in which all colors appear in every face is N P-complete even for graphs in which all faces are of length 3 or 4 only. If all faces are of length 3 this can be decided in polynomial time. The investigation of this problem is motivated by its connection to a variant of the art gallery problem in computational geometry.

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“…Our main result bounds the minimum possible polychromatic number for plane graphs G with g(G) = g. This settles a question raised in [9]. Note that the set 3g− 5 4 , .…”

Section: Introductionsupporting

confidence: 66%

Section: Related Workmentioning

confidence: 99%

Polychromatic colorings of plane graphs

Alon

Berke

Buchin

et al. 2008

Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry

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“…This settles a question raised in [13]. Note that the set { The lower bound for p(G) in (1) remains true when restricting to simple plane graphs G with g(G) = g. Moreover, the constructions for the upper bound in (1) are in fact simple plane graphs.…”

Section: It Is Clear That For Every Plane Graph G P(g(g)) ≤ P(g) ≤ G(g)mentioning

confidence: 74%

“…Horev and Krakovski showed in [13] that any connected plane graph G with g(G) ≥ 3 and maximum degree at most 3, which is not K 4 or a subdivision of K 4 on 5 vertices, are polychromatically 3-colorable. In [7] it is shown that every bipartite cubic plane graph can be colored with 4 colors so that every bounded face of G is polychromatic.…”

Section: Corollary 4 Every Plane Graph G With G(g)mentioning

confidence: 99%

Polychromatic Colorings of Plane Graphs

Alon

Berke

Buchin

et al. 2009

Discrete Comput Geom

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We show that the vertices of any plane graph in which every face is incident to at least g vertices can be colored by (3g − 5)/4 colors so that every color appears in every face. This is nearly tight, as there are plane graphs where all faces are incident to at least g vertices and that admit no vertex coloring of this type with more than (3g + 1)/4 colors. We further show that the problem of determining whether a plane graph admits a vertex coloring by k colors in which all colors appear in every face is in P for k = 2 and is N P-complete for k = 3, 4. We refine this result for polychromatic 3-colorings restricted to 2-connected graphs which have face sizes from a prescribed (possibly infinite) set of integers. Thereby we find an almost complete characterization of these sets of integers (face sizes) for which the corresponding decision problem is in P, and for the others it is N P-complete.

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“…Mohar andŠkrekovski [10] proved using the four-color theorem that every simple plane graph admits a polychromatic 2-coloring, later Bose et al [3] proved that without using the four-color theorem. Horev and Krakovski [9] proved that every plane graph of degree at most 3, other than K 4 admits a polychromatic 3-coloring. Horev et al [7] proved that every 2-connected cubic bipartite plane graph admits a polychromatic 4-coloring.…”

Section: Introductionmentioning

confidence: 99%

Polychromatic Colorings of n-Dimensional Guillotine-Partitions

Keszegh

Lecture Notes in Computer Science

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Abstract. A strong hyperbox-respecting coloring of an n-dimensional hyperbox partition is a coloring of the corners of its hyperboxes with 2 n colors such that any hyperbox has all the colors appearing on its corners. A guillotine-partition is obtained by starting with a single axis-parallel hyperbox and recursively cutting a hyperbox of the partition into two hyperboxes by a hyperplane orthogonal to one of the n axes. We prove that there is a strong hyperbox-respecting coloring of any n-dimensional guillotine-partition. This theorem generalizes the result of Horev et al.[8] who proved the 2-dimensional case. This problem is a special case of the n-dimensional variant of polychromatic colorings. The proof gives an efficient coloring algorithm as well.

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Polychromatic colorings of bounded degree plane graphs

Cited by 15 publications

References 6 publications

Polychromatic colorings of plane graphs

Polychromatic colorings of plane graphs

Polychromatic Colorings of Plane Graphs

Polychromatic Colorings of n-Dimensional Guillotine-Partitions

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