A Proof of a Conjecture of Ohba

Noel, Jonathan A.; Reed, Bruce; Wu, Hehui

doi:10.1002/jgt.21819

Cited by 37 publications

(41 citation statements)

References 32 publications

(51 reference statements)

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“…Therefore,

χ (G_{D}) = k

. Furthermore, it is easy to see that

G_{D}

satisfies the condition stated in Theorem , since:

false| V false(G_{D} false) false| = false| V false(D false) false| \leq 2 \vec{χ} false(D false) + 1 = 2 k + 1 = 2 χ false(G_{D} false) + 1 .

Hence,

χ false(G_{D} false) = χ_{ℓ} false(G_{D} false) = k

since Theorem is verified . Now consider any k ‐list‐assignment L to D .…”

Section: Ohba's Conjecture For Digraphsmentioning

confidence: 90%

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List coloring digraphs

Bensmail

Harutyunyan

2017

Journal of Graph Theory

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The dichromatic number χ⃗false(Dfalse) of a digraph D is the least number k such that the vertex set of D can be partitioned into k parts each of which induces an acyclic subdigraph. Introduced by Neumann‐Lara in 1982, this digraph invariant shares many properties with the usual chromatic number of graphs and can be seen as the natural analog of the graph chromatic number. In this article, we study the list dichromatic number of digraphs, giving evidence that this notion generalizes the list chromatic number of graphs. We first prove that the list dichromatic number and the dichromatic number behave the same in many contexts, such as in small digraphs (by proving a directed version of Ohba's conjecture), tournaments, and random digraphs. We then consider bipartite digraphs, and show that their list dichromatic number can be as large as Ω(prefixlog2n). We finally give a Brooks‐type upper bound on the list dichromatic number of digon‐free digraphs.

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“…Therefore,

χ (G_{D}) = k

. Furthermore, it is easy to see that

G_{D}

satisfies the condition stated in Theorem , since:

false| V false(G_{D} false) false| = false| V false(D false) false| \leq 2 \vec{χ} false(D false) + 1 = 2 k + 1 = 2 χ false(G_{D} false) + 1 .

Hence,

χ false(G_{D} false) = χ_{ℓ} false(G_{D} false) = k

since Theorem is verified . Now consider any k ‐list‐assignment L to D .…”

Section: Ohba's Conjecture For Digraphsmentioning

confidence: 90%

“…In [15], Ohba conjectured that the equality = should hold for every undirected graph with relatively small order (compared to their chromatic number). Ohba's conjecture was recently proved in [22] by Noel, Reed, and Wu. Theorem 2.1 (Ohba's conjecture).…”

Section: Ohba's Conjecture For Digraphsmentioning

confidence: 94%

List coloring digraphs

Bensmail

Harutyunyan

2017

Journal of Graph Theory

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“…The coloring algorithm, which we call Painter, must choose an independent set of marked vertices to receive that color. Colored Ohba [11] conjectured that G is chromatic-choosable when |V (G)| ≤ 2χ(G) + 1; after partial results in [8,11,12,14], this was proved by Noel, Reed, and Wu [10]. Various researchers (see [7]) observed that the complete multipartite graph K 2,...,2,3 is chromatic-choosable but not chromatic-paintable, so the paintability analogue is slightly different: Conjecture 1.2.…”

Section: Introductionmentioning

confidence: 98%

Three topics in online list coloring

Carraher

Loeb

Mahoney

et al. 2014

Journal of Combinatorics

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In online list coloring (introduced by Zhu and by Schauz in 2009), on each round the set of vertices having a particular color in their lists is revealed, and the coloring algorithm chooses an independent subset to receive that color. The paint number of a graph G is the least k such that there is an algorithm to produce a successful coloring with no vertex being shown more than k times; it is at least the choice number. We study paintability of joins with complete or empty graphs, obtaining a partial result toward the paint analogue of Ohba's Conjecture. We also determine upper and lower bounds on the paint number of complete bipartite graphs and characterize 3-paintcritical graphs.

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“…Quite recently, Ohba's conjecture has been verified by Noel et al [6] with a complicated proof. However, it seems difficult to apply their method to handle the conjecture of Zhen and Wu mentioned above.…”

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confidence: 94%