Colorings of plane graphs: A survey

Borodin, O. V.

doi:10.1016/j.disc.2012.11.011

Cited by 155 publications

(187 citation statements)

References 145 publications

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“…The main problem if planar graphs are (3,1)-choosable remains open. We hope that this paper could serve as an inspiration of possible approaches to the problem.…”

Section: Resultsmentioning

confidence: 99%

On Choosability with Separation of Planar Graphs with Forbidden Cycles

Choi

Lidický

Stolee

2015

Journal of Graph Theory

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We study choosability with separation which is a constrained version of list coloring of graphs. A (k, d)-list assignment L of a graph G is a function that assigns to each vertex v a list L(v) of at least k colors and for any adjacent pair xy, the lists L(x) andThis concept is also known as choosability with separation. We prove that planar graphs without 4-cycles are (3, 1)-choosable and that planar graphs without 5-cycles and 6-cycles are (3, 1)-choosable. In addition, we give an alternative and slightly stronger proof that triangle-free planar graphs are (3, 1)-choosable.

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“…The main problem if planar graphs are (3,1)-choosable remains open. We hope that this paper could serve as an inspiration of possible approaches to the problem.…”

Section: Resultsmentioning

confidence: 99%

On Choosability with Separation of Planar Graphs with Forbidden Cycles

Choi

Lidický

Stolee

2015

Journal of Graph Theory

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“…For other information on the 3-colorability of planar graphs, we refer the reader to the excellent survey [3] by Borodin. Conjectures 1 and 2 seem very hard to confirm.…”

Section: Conjecture 2 Every Planar Graph With D ∇ ≥ 4 Is 3-colorablementioning

confidence: 99%

Distance Constraints on Short Cycles for 3-Colorability of Planar graphs

Kang

Wang

2014

Graphs and Combinatorics

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Montassier et al. showed that every planar graph without cycles of length at most five at distance less than four is 3-colorable [A relaxation of of Havel's 3-color problem, Inform. Process. Lett. 107 (2008) 107-109]. Borodin, Montassier and Raspaud asked in [Planar graphs without adjacent cycles of length at most seven are 3-colorable, Discrete Math. 310 (2010) 167-173]: is every planar graph without adjacent cycles of length at most five 3-colorable? In this note, we show that every planar graph without cycles of length at most five at distance less than two is 3-colorable.

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“…Some extensions of Kotzig's Theorem and their application to colouring which was an important stimulating factor for extending Kotzig's Theorem, are discussed in [4].…”

Section: Introductionmentioning

confidence: 99%

“…Every 3-connected plane graph has a 3-path of one of the following: types: (10, 3, 10), (7,4,7), (6,5,6), (3,4,15), (3,6,11), (3,8,5), (3,10,3), (4,4,11), (4,5,7), or (4,7,5).…”

Section: Introductionmentioning

confidence: 99%

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Note on 3-paths in plane graphs of girth 4

Jendroľ

Maceková

Soták

2015

Discrete Mathematics

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a b s t r a c t An (i, j, k)-path is a path on three vertices u, v and w in this order with deg(u) ≤ i, deg(v) ≤ j, and deg(w) ≤ k. In this paper, we prove that every connected plane graph of girth 4 and minimum degree at least 2 has at least one of the following: a (2, ∞, 2)-path, a (2, 7, 3)-path, a (3, 5, 3)-path, a (4, 2, 5)-path, or a (4, 3, 4)-path. Moreover, no parameter of this description can be improved. Our result supplements recent results concerning the existence of specific 3-paths in plane graphs.

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Colorings of plane graphs: A survey

Cited by 155 publications

References 145 publications

On Choosability with Separation of Planar Graphs with Forbidden Cycles

On Choosability with Separation of Planar Graphs with Forbidden Cycles

Distance Constraints on Short Cycles for 3-Colorability of Planar graphs

Note on 3-paths in plane graphs of girth 4

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