Ohba’s conjecture is true for graphs with independence number at most three

Shen, Yufa; He, Wei; Zheng, Guoping; Yan-po, LI

doi:10.1016/j.aml.2009.01.001

Cited by 12 publications

(8 citation statements)

References 8 publications

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“…Chromatic choosable graphs with independence number 3. Ohba Conjecture was proved to be true for graphs with independence number at most 3 already in [16]. We present an alternative proof based on Lemma 5.…”

Section: 1mentioning

confidence: 90%

Towards an on-line version of Ohba’s conjecture

Kozik

Micek

Zhu

2014

European Journal of Combinatorics

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The on-line choice number of a graph is a variation of the choice number defined through a two person game. It is at least as large as the choice number for all graphs and is strictly larger for some graphs. In particular, there are graphs G with |V (G)| = 2χ(G) + 1 whose on-line choice numbers are larger than their chromatic numbers, in contrast to a recently confirmed conjecture of Ohba that every graph G with |V (G)| 2χ(G) + 1 has its choice number equal its chromatic number. Nevertheless, an on-line version of Ohba conjecture was proposed in [P. Huang, T. Wong and X. Zhu, Application of polynomial method to on-line colouring of graphs, European J. Combin., 2011]: Every graph G with |V (G)| 2χ(G) has its on-line choice number equal its chromatic number. This paper confirms the on-line version of Ohba conjecture for graphs G with independence number at most 3. We also study list colouring of complete multipartite graphs K 3⋆k with all parts of size 3. We prove that the on-line choice number of K 3⋆k is at most 3 2 k, and present an alternate proof of Kierstead's result that its choice number is ⌈(4k − 1)/3⌉. For general graphs G, we prove that if |V (G)| χ(G) + χ(G) then its on-line choice number equals chromatic number.

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Section: 1mentioning

confidence: 90%

Towards an on-line version of Ohba’s conjecture

Kozik

Micek

Zhu

2014

European Journal of Combinatorics

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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. Part (iv) was proved by He et al [5].…”

Section: Introductionmentioning

confidence: 65%

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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“…An early result in this direction is due to Ohba [Ohb04], who proved that if |V (G)| ≤ 2χ(G) and α(G) ≤ 3, then G is chromatic-choosable. By building on Ohba's techniques, He, Li, Shen and Zheng [SHZL09] extended this result to graphs of order 2χ + 1. Their technique can be roughly outlined as follows.…”

Section: Partial Resultsmentioning

confidence: 99%