Partitioning H -minor free graphs into three subgraphs with no large components

Liu, Chun Hung; Oum, Sang Il

doi:10.1016/j.endm.2015.06.020

Cited by 15 publications

(28 citation statements)

References 18 publications

(20 reference statements)

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“…They also proved that the clustered chromatic number of the class of graphs with no K t minor is at most 4t − 4. Liu and Oum [22] proved that for every graph H, every graph G with no H minor and maximum degree at most is 3-colourable with clustering f (H, ) for some function f , which generalises the result of Esperet and Joret [7] for graphs embeddable on surfaces of bounded Euler genus. Combined with Theorem 2.1, this implies that the clustered chromatic number of the class of graphs with no K t minor is at most 3t − 3.…”

Section: Previous Results On Improper Colouring and Forbidden Minorsmentioning

confidence: 58%

Improper colouring of graphs with no odd clique minor

Kang

Oum

2019

Combinator. Probab. Comp.

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As a strengthening of Hadwiger’s conjecture, Gerards and Seymour conjectured that every graph with no odd Kt minor is (t − 1)-colourable. We prove two weaker variants of this conjecture. Firstly, we show that for each t ⩾ 2, every graph with no odd Kt minor has a partition of its vertex set into 6t − 9 sets V1, …, V6t−9 such that each Vi induces a subgraph of bounded maximum degree. Secondly, we prove that for each t ⩾ 2, every graph with no odd Kt minor has a partition of its vertex set into 10t −13 sets V1,…, V10t−13 such that each Vi induces a subgraph with components of bounded size. The second theorem improves a result of Kawarabayashi (2008), which states that the vertex set can be partitioned into 496t such sets.

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Section: Previous Results On Improper Colouring and Forbidden Minorsmentioning

confidence: 58%

Improper colouring of graphs with no odd clique minor

Kang

Oum

2019

Combinator. Probab. Comp.

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“…In particular, they proved that every K t -minor-free graph is 31 2 t -colourable with clustering f (t), for some function f . The number of colours in this result was improved to 1 2 (7t − 3) by Wood [52] ‡ , to 4t − 4 by Edwards et al [16], and to 3t − 3 by Liu and Oum [38]. See [28,29] for analogous results for graphs excluding odd minors.…”

Section: Introductionmentioning

confidence: 75%

“…We leave as an open problem to determine the clustered chromatic number of graphs excluding a (strong or weak) K t immersion. It was observed by both Haxell et al [26] and Liu and Oum [38] that the results in Alon et al [2] prove that for every k, N , there exists a (4k − 2)-regular graph G such that every k-colouring of G has a monochromatic component of size at least N . In other words, the clustered chromatic number of graphs with maximum degree Δ is at least 1 4 (Δ + 6) .…”

Section: Excluded Immersionsmentioning

confidence: 98%

“…Proof. Liu and Oum [38] proved that for every minor-closed graph class G and integer d, there is an integer c = c(G, d) such that every graph in G with maximum degree d is 3-colourable with clustering c. (Esperet and Joret [17] previously proved an analogous result for graphs on surfaces.) Let k be the defective chromatic number of G. Thus for some integer d, every graph G in G is k-colourable with defect d. Apply the result of Liu and Oum [38] to each monochromatic component of G, which has maximum degree at most d. Then G is 3k-colourable with clustering c, and hence the clustered chromatic number of G is at most 3k.…”

Section: Introductionmentioning

confidence: 95%

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Improper colourings inspired by Hadwiger's conjecture

Heuvel

Wood

2018

J. London Math. Soc.

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Hadwiger's conjecture asserts that every Kt‐minor‐free graph has a proper (t−1)‐colouring. We relax the conclusion in Hadwiger's conjecture via improper colourings. We prove that every Kt‐minor‐free graph is (2t−2)‐colourable with monochromatic components of order at most ⌈0false12(t−2)⌉. This result has no more colours and much smaller monochromatic components than all previous results in this direction. We then prove that every Kt‐minor‐free graph is (t−1)‐colourable with monochromatic degree at most t−2. This is the best known degree bound for such a result. Both these theorems are based on a decomposition method of independent interest. We give analogous results for Ks,t‐minor‐free graphs, which lead to improved bounds on generalised colouring numbers for these classes. Finally, we prove that graphs containing no Kt‐immersion are 2‐colourable with bounded monochromatic degree.

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“…Both the t-component chromatic number [1,2,3,15,20,21,27,32,33,34,35,36,38] and the t-component stability number [14,16,26,29,44] have been actively considered from several viewpoints, especially in graph theory and theoretical computer science.…”

Section: Further Remarksmentioning

confidence: 99%

Bounded monochromatic components for random graphs

Broutin

Kang

2018

Journal of Combinatorics

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We consider vertex partitions of the binomial random graph Gn,p. For np → ∞, we observe the following phenomenon: in any partition into asymptotically fewer than χ(Gn,p) parts, i.e. o(np/ log np) parts, one part must induce a connected component of order at least roughly the average part size.Stated another way, we consider the t-component chromatic number, the smallest number of colours needed in a colouring of the vertices for which no monochromatic component has more than t vertices. As long as np → ∞, there is a threshold for t around Θ(p −1 log np): if t is smaller then the t-component chromatic number is nearly as large as the chromatic number, while if t is greater then it is around n/t.For 0 < p < 1 fixed, we obtain more precise information. We find something more subtle happens at the threshold t = Θ(log n), and we determine that the asymptotic first-order behaviour is characterised by a non-smooth function. Moreover, we consider the t-component stability number, the maximum order of a vertex subset that induces a subgraph with maximum component order at most t, and show that it is concentrated in a constant length interval about an explicitly given formula, so long as t = O(log log n).We also consider a related Ramsey-type parameter and use bounds on the component stability number of G n,1/2 to describe its basic asymptotic growth.

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Partitioning H -minor free graphs into three subgraphs with no large components

Cited by 15 publications

References 18 publications

Improper colouring of graphs with no odd clique minor

Improper colouring of graphs with no odd clique minor

Improper colourings inspired by Hadwiger's conjecture

Bounded monochromatic components for random graphs

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