Planar Graphs without 7-Cycles Are 4-Choosable

Farzad, Babak

doi:10.1137/05064477x

Cited by 33 publications

(29 citation statements)

References 9 publications

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“…Fijavž et al [6] proved that planar graphs without 6-cycles are 4-choosable. Farzad [5] showed that a planar graph without 7-cycles is 4-choosable. Xu [23], and Wang and Lih [21] independently, proved that planar graphs without two triangles sharing a common vertex are 4-choosable.…”

Section: Introductionmentioning

confidence: 99%

RETRACTED ARTICLE: On 3-choosability of planar graphs with neither adjacent triangles nor 5-, 6- and 9-cycles

Zhang

2010

Information Processing Letters

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Section: Introductionmentioning

confidence: 99%

RETRACTED ARTICLE: On 3-choosability of planar graphs with neither adjacent triangles nor 5-, 6- and 9-cycles

Zhang

2010

Information Processing Letters

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“…Hence none of u 1 , u 3 , u 4 can be a special vertex, and thus there is only at most one special vertex. 4 be the neighbors of v in cyclic order so that vu 1 u 2 is the 3-face f 1 incident to v. Let the 4-face f 2 be u 2 vu 3 x and let the other 4-face f 4 be u 1 vu 4 y. If x = u 1 , then d(u 2 ) = 2, which is a contradiction.…”

Section: Lemma 22 V (G) Does Not Contain Any Of the Followingmentioning

confidence: 99%

Toroidal graphs containing neither nor 6‐cycles are 4‐choosable

Choi

2016

Journal of Graph Theory

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The choosability χℓfalse(Gfalse) of a graph G is the minimum k such that having k colors available at each vertex guarantees a proper coloring. Given a toroidal graph G, it is known that χℓfalse(Gfalse)≤7, and χℓfalse(Gfalse)=7 if and only if G contains K7. Cai et al. (J Graph Theory 65(1) (2010), 1–15) proved that a toroidal graph G without 7‐cycles is 6‐choosable, and χℓfalse(Gfalse)=6 if and only if G contains K6. They also proved that a toroidal graph G without 6‐cycles is 5‐choosable, and conjectured that χℓfalse(Gfalse)=5 if and only if G contains K5. We disprove this conjecture by constructing an infinite family of non‐4‐colorable toroidal graphs with neither K5 nor cycles of length at least 6; moreover, this family of graphs is embeddable on every surface except the plane and the projective plane. Instead, we prove the following slightly weaker statement suggested by Zhu: toroidal graphs containing neither K5− (a K5 missing one edge) nor 6‐cycles are 4‐choosable. This is sharp in the sense that forbidding only one of the two structures does not ensure that the graph is 4‐choosable.

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“…Is every (3, 1)-choosable graph (4, 0)-choosable? It is known that every planar graph without k-cycles for some k ∈ {3, 4, 5, 6, 7} is (4, 0)-choosable [10].…”

Section: Dischargingmentioning

confidence: 99%

Improper Choosability of Planar Graphs without 4-Cycles

Wang¹,

Xu²

2013

SIAM J. Discrete Math.

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Let k ≥ 2 be a positive integer, and let d be a nonnegative integer. A graph G is d-improperly k-choosable, or simply, (k, d)-choosable if, for every list assignment L with |L(v)| ≥ k for every v ∈ V (G), there exists an L-coloring of G such that each vertex of G has at most d neighbors colored the same color as itself. It is known that every planar graph with cycles of length neither 4 nor k for some k ∈ {5, 6, 7, 8, 9} is (3, 1)-choosable. In this paper, we prove that every planar graph without cycles of length 4 is (3, 1)-choosable. This is best possible in the following two senses: (i) there are planar graphs which are not (3, 1)-colorable, hence not (3, 1)-choosable; (ii) there are planar graphs without 4-cycles which are not (3, 0)-colorable, hence not (3, 0)-choosable. Introduction.All graphs considered in this paper are finite, simple, and undirected. For the used but undefined terminology and notation, we refer the reader to the book by Bondy and Murrty [3].Let G = (V, E) be a graph with the set of vertices V and the set of edgesThe classical colorability of a graph G now has various natural generalizations in the literature. Two of them are the so-called (k, d)-colorability, and list colorability or choosability.Let k ≥ 2 be a positive integer, and let d be a nonnegative integer. A graph G = (V, E) is (k, d)-colorable if G can be colored using k colors such that every vertex has at most d-neighbors colored the same as itself. In terms of (k, d)-colorability, some famous theorems and conjectures in the literature may be listed as follows:(1) Four color theorem [1, 2]: Every planar graph is (4, 0)-colorable.(2) Three color theorem [11]: Every triangle-free planar graph is (3, 0)-colorable.(3) Steinberg's conjecture [16]: Every planar graph with cycles of length neither 4 nor 5 is (3, 0)-colorable.(4) Bordeaux conjecture [4]: Every planar graph with neither adjacent triangles nor cycles of length 5 is (3, 0)-colorable.Below are two interesting known results on (k, d)-colorability of planar graphs.(5) The (3, 2)-colorability of planar graphs [5]: Every planar graph is (3, 2)-colorable.(6) A partial solution to Bordeaux conjecture [20]: Every planar graph with neither adjacent triangles nor cycles of length 5 is (3, 1)-colorable.As pointed out by the author of [20], (6) implies that if G is a planar graph with neither adjacent triangles nor cycles of length 5, then G has a matching M such that G − M is (3, 0)-colorable, thus providing a partial solution to Bordeaux conjecture.

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Planar Graphs without 7-Cycles Are 4-Choosable

Cited by 33 publications

References 9 publications

RETRACTED ARTICLE: On 3-choosability of planar graphs with neither adjacent triangles nor 5-, 6- and 9-cycles

RETRACTED ARTICLE: On 3-choosability of planar graphs with neither adjacent triangles nor 5-, 6- and 9-cycles

Toroidal graphs containing neither nor 6‐cycles are 4‐choosable

Improper Choosability of Planar Graphs without 4-Cycles

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