A note on acyclic vertex-colorings

Sereni, Jean-Sébastien; Volec, Jan

doi:10.4310/joc.2016.v7.n4.a8

Cited by 9 publications

(7 citation statements)

References 15 publications

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“…(log ∆) 4/3 . Recently, the upper bound was improved to 6.59∆ 4 3 + 3.3∆ by Ndreca et al [30] and then to 2.835∆ 4 3 + ∆ by Sereni and Volec [34].…”

Section: Acyclic Coloring Of Graphsmentioning

confidence: 99%

Entropy compression method applied to graph colorings

Gonçalves,

Montassier,

Pinlou

2014

Preprint

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Based on the algorithmic proof of Lovász local lemma due to Moser and Tardos, the works of Grytczuk et al. on words, and Dujmović et al. on colorings, Esperet and Parreau developed a framework to prove upper bounds for several chromatic numbers (in particular acyclic chromatic index, star chromatic number and Thue chromatic number) using the so-called entropy compression method.Inspired by this work, we propose a more general framework and a better analysis. This leads to improved upper bounds on chromatic numbers and indices. In particular, every graph with maximum degree ∆ has an acyclic chromatic number at most 3 2 ∆ 4 3 + O(∆). Also every planar graph with maximum degree ∆ has a facial Thue choice number at most ∆ + O(∆ 1 2 ) and facial Thue choice index at most 10. * This research is partially supported by the ANR EGOS, under contract ANR-12-JS02-002-01. The algorithmLet V ∈ {1, 2, . . . , κ} t be a vector of length t, for some arbitrarily large t ≫ n = |V (G)|. Algorithm ACYCLICCOLORINGGAMMA_G (see below) takes the vector V as input and returns a partial acyclic coloring ϕ : V (G) → {•, 1, 2, . . . , κ} of G (• means that the vertex is uncolored) and a text file R that is called a record in the remaining of the paper. The acyclic coloring ϕ is necessarily partial since we try to color G with a number of colors less than its acyclic chromatic number. For a given vertex v of G, we denote by N (v) the set of neighbors of v.Algorithm 1: ACYCLICCOLORINGGAMMA_G Input : V (vector of length t).Output: (ϕ, R).Write "Uncolor, neighbor u \n" in R 11 else if v belongs to a bicolored cycle of length 2k (k ≥ 2), say (v = u 1 , . . . , u 2k ) then // Bicolored cycle issue 12 for j ← 1 to 2k − 2 do 13 ϕ(u j ) ← • 14 Write "Uncolor, 2k-cycle (v = u 1 , . . . , u 2k ) \n" in R 15 return (ϕ, R)

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“…(log ∆) 4/3 . Recently, the upper bound was improved to 6.59∆ 4 3 + 3.3∆ by Ndreca et al [30] and then to 2.835∆ 4 3 + ∆ by Sereni and Volec [34].…”

Section: Acyclic Coloring Of Graphsmentioning

confidence: 99%

Entropy compression method applied to graph colorings

Gonçalves,

Montassier,

Pinlou

2014

Preprint

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“…Esperet and Parreau suggested, through further examples and applications, that their algorithm could be adapted to treat most of the applications in graph coloring problems covered by the LLL. Indeed, this was confirmed in several successive papers [26,42,39,43,15,14,45,24,8], where the Esperet-Parreau scheme has been applied to various graph coloring problems and beyond, generally improving previous results obtained via the LLL/CELL (sometimes the improvement is more sensible, sometimes less). However, in all papers mentioned above the Esperet-Parreau algorithmic scheme, usually called entropy compression method (the name is probably due to Tao [48]), has been commonly utilized as a set of ad hoc instructions to be implemented on a case-by-case basis.…”

Section: The Entropy Compression Methodsmentioning

confidence: 60%

Moser-Tardos resampling algorithm, entropy compression method and the subset gas

Fialho¹,

Lima²,

Procacci³

2020

Preprint

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“…This construction is nearly best possible, since they also found a constant C 2 such that χ a (G) ≤ C 2 ∆ 4/3 for every graph G with maximum degree ∆. The best known upper bound is χ a (G) ≤ 2.835∆ 4/3 + ∆, due to Sereni and Volec [14]. Now we turn to acyclic edge-coloring.…”

Section: Introductionmentioning

confidence: 97%

Acyclic Edge-Coloring of Planar Graphs: $\Delta$ Colors Suffice When $\Delta$ is Large

Cranston¹

2019

SIAM J. Discrete Math.

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An acyclic edge-coloring of a graph G is a proper edge-coloring of G such that the subgraph induced by any two color classes is acyclic. The acyclic chromatic index, χ ′ a (G), is the smallest number of colors allowing an acyclic edge-coloring of G. Clearly χ ′ a (G) ≥ ∆(G) for every graph G. Cohen, Havet, and Müller conjectured that there exists a constant M such that every planar graph with ∆(G) ≥ M has χ ′ a (G) = ∆(G). We prove this conjecture.

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A note on acyclic vertex-colorings

Abstract: We prove that the acyclic chromatic number of a graph with maximum degree ∆ is less than 2.835∆ 4/3 + ∆. This improves the previous upper bound, which was 50∆ 4/3 . To do so, we draw inspiration from works by Alon, McDiarmid and Reed and by Esperet and Parreau.

Cited by 9 publications

References 15 publications

Entropy compression method applied to graph colorings

Entropy compression method applied to graph colorings

Moser-Tardos resampling algorithm, entropy compression method and the subset gas

Acyclic Edge-Coloring of Planar Graphs: $\Delta$ Colors Suffice When $\Delta$ is Large

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