On choosability of some complete multipartite graphs and Ohba's conjecture

Shen, Yufa; He, Wei; Zheng, Guoping; Wang, Yanning; Zhang, Lingmin

doi:10.1016/j.disc.2007.03.059

Cited by 10 publications

(6 citation statements)

References 7 publications

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“…Adding one dominant vertex to K 3,3 gives K 1,3,3 , which again is 3-choosable [16]. In fact, this property holds for infinitely many complete bipartite graphs [16]; also see [19].…”

mentioning

confidence: 99%

Disproof of the List Hadwiger Conjecture

Barát¹,

Joret²,

Wood³

2011

Electron. J. Combin.

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The List Hadwiger Conjecture asserts that every K t -minor-free graph is t-choosable. We disprove this conjecture by constructing a K 3t+2 -minor-free graph that is not 4tchoosable for every integer t ≥ 1. IntroductionIn 1943, Hadwiger [6] made the following conjecture, which is widely considered to be one of the most important open problems in graph theory; see [26] for a survey 1 .The Hadwiger Conjecture holds for t ≤ 6 (see [3,6,17,18,28]) and is open for t ≥ 7. In fact, the following more general conjecture is open.Weak Hadwiger Conjecture. Every K t -minor-free graph is ct-colourable for some constant c ≥ 1.It is natural to consider analogous conjectures for list colourings 2 . First, consider the * MSC: graph minors 05C83, graph coloring 05C15 See [2] for undefined graph-theoretic terminology. Let [a, b] := {a, a + 1, . . . , b}. 2 A list-assignment of a graph G is a function L that assigns to each vertex v of G a set L(v) of colours.

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“…Adding one dominant vertex to K 3,3 gives K 1,3,3 , which again is 3-choosable [16]. In fact, this property holds for infinitely many complete bipartite graphs [16]; also see [19].…”

mentioning

confidence: 99%

Disproof of the List Hadwiger Conjecture

Barát¹,

Joret²,

Wood³

2011

Electron. J. Combin.

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“…Part (ii) of Theorem C follows from part (i) if t = 0 and from Theorem B(iv) if t = 1; it was proved by Shen et al [11] for t = 2, 3, and by Shen et al [12] for t = 4. Part (iii) is the same as part (i) or part (ii) if t = 0 or 1, respectively; it was proved by He et al [5] for t = 2, and by Shen et al [10] in general.…”

Section: Introductionmentioning

confidence: 80%

Ohba’s conjecture for graphs with independence number five

Kostochka

Stiebitz

Woodall

2011

Discrete Mathematics

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a b s t r a c tOhba has conjectured that if G is a k-chromatic graph with at most 2k + 1 vertices, then the list chromatic number or choosability ch(G) of G is equal to its chromatic number χ (G), which is k. It is known that this holds if G has independence number at most three. It is proved here that it holds if G has independence number at most five. In particular, and equivalently, it holds if G is a complete k-partite graph and each part has at most five vertices.

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“…Ohba [16] conjectured that for any positive integer k, k-colourable graphs with at most 2k +1 vertices are k-choosable. This conjecture has been studied in many papers [10,12,[14][15][16][17][19][20][21], and was confirmed by Noel, Reed and Wu [15]: Theorem 1.1 Every k-colourable graph with at most 2k + 1 vertices is k-choosable.…”

Section: Introductionmentioning

confidence: 89%

“…Theorem 4.1 takes care of complete k-partite graphs with 3 − -parts, except that either one 5-part or at most two 4-parts. It was proved in [21] that G = K 6,2⋆(k−3),1⋆2 is kchoosable. So either G has a 6 + -part and G ≠ K 6,2⋆(k−3),1⋆2 , or two 5-parts or three 4-parts.…”

Section: Pseudo-l-colouringmentioning

confidence: 99%

Minimum non-chromatic-choosable graphs with given chromatic number

Zhu¹,

Zhu²

2022

Preprint

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A graph G is called chromatic-choosable if χ(G) = ch(G). A natural problem is to determine the minimum number of vertices in a k-chromatic non-k-choosable graph. It was conjectured by Ohba, and proved by Noel, Reed and Wu thatand G = K 4,2⋆(k−1) are k-chromatic graphs with V (G) = 2k + 2 that are not k-choosable. Some subgraphs of these two graphs are also non-k-choosable. The main result of this paper is that all other k-chromatic graphs G with V (G) = 2k+2 are k-choosable. In particular, if χ(G) is odd and V (G) ≤ 2χ(G) + 2, then G is chromatic-choosable, which was conjectured by Noel.

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On choosability of some complete multipartite graphs and Ohba's conjecture

Cited by 10 publications

References 7 publications

Disproof of the List Hadwiger Conjecture

Disproof of the List Hadwiger Conjecture

Ohba’s conjecture for graphs with independence number five

Minimum non-chromatic-choosable graphs with given chromatic number

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