On 3-colorability of planar graphs without adjacent short cycles

Wang, Yingqian; Mao, XiangHua; Lu, Huajing; Wang, Weifan

doi:10.1007/s11425-010-0074-y

Cited by 10 publications

(4 citation statements)

References 7 publications

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“…Another relaxation of Nsk3CC is proved by Borodin, Glebov and Raspaud in [14]: every planar graph without triangular cycles of length from 4 to 7 (or, which is equivalent, without triangular cycles of length in {3, 5, 7} or {4, 5, 7}) is 3-colorable. In particular, this implies the 3-colorability of planar graphs having no cycle with the length belonging to any of the sets {4, 5, 7}, {4, 6, 7} or {4, 6, 8} and absorbs the results in [13,15,16,18,19,27,[30][31][32].…”

Section: Conjecture 3 Every Planar Graph With D ∇ 4 and Without Triamentioning

confidence: 83%

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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Section: Conjecture 3 Every Planar Graph With D ∇ 4 and Without Triamentioning

confidence: 83%

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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“…The best known bound for k is 7 [9]. A cycle C is triangular if it is adjacent to a triangle other than C. In [6], it is proved that every planar graph without triangular cycles of length from 4 to 7 is 3-colorable, which implies all results in [7,8,9,10,11,21,26,27,28,29].…”

Section: Theorem 6 ([3]mentioning

confidence: 99%

Short proofs of coloring theorems on planar graphs

Borodin

Kostochka

Lidický

et al. 2014

European Journal of Combinatorics

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A recent lower bound on the number of edges in a k-critical n-vertex graph by Kostochka and Yancey yields a half-page proof of the celebrated Gr\"otzsch Theorem that every planar triangle-free graph is 3-colorable. In this paper we use the same bound to give short proofs of other known theorems on 3-coloring of planar graphs, among whose is the Gr\"unbaum-Aksenov Theorem that every planar with at most three triangles is 3-colorable. We also prove the new result that every graph obtained from a triangle-free planar graph by adding a vertex of degree at most four is 3-colorable.Comment: 13 pages, 4 figure

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“…有兴趣的读者可参见文献 [14] 以了解更多与 χ-有界的问题相关的结果. 另外, 读者也可参见文献 [15][16][17][18][19][20] 了解更多关于曲面、图、以及染色的问题、方法与结果.…”

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A note on induced subtrees and chromatic number of graphs

Zhang¹,

Xu²

2016

Sci. Sin.-Math.

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Gyárfás 等 (1980) 证明了, 若一个图 G 不含三角形和长为 4 的圈, 则 G 含有任一个 χ(G) 个顶点的树作为诱导子图. 另外, 他们还证明了, 若 G 不含三角形, 且 χ(G) m + n, 则 G 一定包含一个特殊的树 (m, n)-mop 作为诱导子图. 本文推广了 Gyárfás 等 (1980) 的这两个结果, 证明了 (1) 若图 G 的任一个顶点至多含在 k 个三角形和 l 个长为 4 的圈中, 且 χ(G) t + 2k + 2l, 则 G 包含任一个 t 个点的树作为诱导子图; (2) 若图 G 中的每一个顶点至多包含在 k 个三角形中, 且不能够诱导出 T , 则 χ(G) < m(k + 1) + n, 其中 T 为 (m, n)-mop.

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On 3-colorability of planar graphs without adjacent short cycles

Abstract: A short cycle means a cycle of length at most 7. In this paper, we prove that planar graphs without adjacent short cycles are 3-colorable. This improves a result of Borodin et al. (2005).

Cited by 10 publications

References 7 publications

A step towards the strong version of Havelʼs three color conjecture

A step towards the strong version of Havelʼs three color conjecture

Short proofs of coloring theorems on planar graphs

A note on induced subtrees and chromatic number of graphs

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