Some counterexamples associated with the three-color problem

Aksionov, V. A.; Mel'nikov, L. S.

doi:10.1016/0095-8956(80)90051-9

Cited by 29 publications

(45 citation statements)

References 2 publications

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“…Nevertheless, we can show that ch * (x) 0 for all x ∈ V (G)∪ F (G). Using (1), this leads to the following obvious contradiction:…”

Section: Claim 2 In the Counterexample Gmentioning

confidence: 82%

“…In 1970, Havel [3] asked if there exists d such that every planar graph with the minimum distance between triangles at least d is 3-colorable. In 1976, Aksionov and Mel'nikov [1] and, independently, Steinberg (see [1]) proved that d 4.…”

Section: Introductionmentioning

confidence: 99%

“…We recall that Aksionov and Mel'nikov [1] proved that if d 3 exists, then d 3 4, and conjectured that 0020-0190/$ -see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2008.02.003 Table 1 Every planar graph with the distance-…”

Section: Introductionmentioning

confidence: 99%

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A relaxation of Havel's 3-color problem

Montassier

Raspaud

Wang

et al. 2008

Information Processing Letters

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“…Nevertheless, we can show that ch * (x) 0 for all x ∈ V (G)∪ F (G). Using (1), this leads to the following obvious contradiction:…”

Section: Claim 2 In the Counterexample Gmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

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A relaxation of Havel's 3-color problem

Montassier

Raspaud

Wang

et al. 2008

Information Processing Letters

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“…It is clear that the graph G is planar and it was proved in [2] that G is 4-chromatic. Now we show that the cover degeneracy LðGÞ ¼ 3.…”

Section: Resultsmentioning

confidence: 97%

A note on the cover degeneracy of graphs

Zhang¹,

Wu²

2005

Journal of Graph Theory

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“…There are 4-chromatic planar graphs with d ∇ = 1 and d ∇ = 2 (Havel [24,25]) and d ∇ = 3 (Aksenov and Mel'nikov [6], modifying Havel's constructions, and…”

Section: Introductionmentioning

confidence: 98%

A step towards the strong version of Havelʼs three color conjecture

Borodin

Glebov

Jensen

2012

Journal of Combinatorial Theory, Series B

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In 1970, Havel asked if each planar graph with the minimum distance, d ∇ , between triangles large enough is 3-colorable. There are 4-chromatic planar graphs with d ∇ = 3 (Aksenov, Mel'nikov, and Steinberg, 1980). The first result in the positive direction of Havel's problem was made in 2003 by Borodin and Raspaud, who proved that every planar graph with d ∇ 4 and no 5-cycles is 3-colorable.Recently, Havel's problem was solved by Dvořák, Král' and Thomas in the positive, which means that there exists a constant d such that each planar graph with d ∇ d is 3-colorable. (As far as we can judge, this d is very large.) We conjecture that the strongest possible version of Havel's problem (SVHP) is true: every planar graph with d ∇ 4 is 3-colorable. In this paper we prove that each planar graph with d ∇ 4 and without 5-cycles adjacent to triangles is 3-colorable. The readers are invited to prove a stronger theorem: every planar graph with d ∇ 4 and without 4-cycles adjacent to triangles is 3-colorable, which could possibly open way to proving SVHP.

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Some counterexamples associated with the three-color problem

Cited by 29 publications

References 2 publications

A relaxation of Havel's 3-color problem

A relaxation of Havel's 3-color problem

A note on the cover degeneracy of graphs

A step towards the strong version of Havelʼs three color conjecture

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