Acyclic edge colorings of graphs

Alon, Noga; Sudakov, Benny; Zaks, Ayal

doi:10.1002/jgt.1010

Cited by 138 publications

(178 citation statements)

References 11 publications

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“…A related result was obtained by Nesetril and Wormald [69] who prove that the acyclic edge chromatic number of a random r-regular graph is a.a.s. equal to +1, improving a previous result of Alon, Sudakov, and Zaks [8].…”

Section: Sketch Of Proofsupporting

confidence: 81%

The cook-book approach to the differential equation method

Dı́az¹,

Mitsche

2010

Computer Science Review

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We survey several applications of the differential equation method in different areas of discrete mathematics. We give examples of its use in the analysis of algorithms in random graph processes and random boolean formulae. We also briefly review the basic theorem of Wormald [77] used in the analysis, but we aim for simplicity and not for maximal generality. The primary goal of this survey is to be a toolbox for the usage of the differential equation method.

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Section: Sketch Of Proofsupporting

confidence: 81%

The cook-book approach to the differential equation method

Dı́az¹,

Mitsche

2010

Computer Science Review

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“…It has been conjectured by Alon, Sudakov and Zaks [2] that a ′ (G) ≤ ∆ + 2 for any G. We were informed by Alon that the same conjecture was raised earlier by Fiamcik [5]. Using probabilistic arguments Alon, McDiarmid and Reed [1] proved that a ′ (G) ≤ 60∆.…”

mentioning

confidence: 70%

“…Our Result: Alon, Sudakov and Zaks [2] suggested a possibility that complete graphs of even order are the only regular graphs which require ∆ + 2 colors to be acyclically edge colored. Nešetřil and Wormald [8] supported the statement by showing that the acyclic edge chromatic number of a random d-regular graph is asymptotically almost surely equal to d + 1 (when d ≥ 2).…”

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confidence: 80%

d‐Regular graphs of acyclic chromatic index at least d+2

Basavaraju

Chandran

Kummini

2009

Journal of Graph Theory

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An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a ′ (G). It was conjectured by Alon, Sudakov and Zaks (and earlier by Fiamcik) that a ′ (G) ≤ ∆ + 2, where ∆ = ∆(G) denotes the maximum degree of the graph. Alon et.al also raised the question whether the complete graphs of even order are the only regular graphs which require ∆ + 2 colors to be acyclically edge colored. In this paper, using a simple counting argument we observe not only that this is not true, but infact all d-regular graphs with 2n vertices and d > n, requires at least d + 2 colors. We also show that a ′ (Kn,n) ≥ n + 2, when n is odd using a more non-trivial argument(Here Kn,n denotes the complete bipartite graph with n vertices on each side). This lower bound for Kn,n can be shown to be tight for some families of complete bipartite graphs and for small values of n. We also infer that for every d, n such that d ≥ 5, n ≥ 2d + 3 and dn even, there exist d-regular graphs which require at least d + 2-colors to be acyclically edge colored.

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“…Alon et al [2] proved that there exists a constant k such that a (G) ≤ +2 for any graph G whose girth is at least k log . They also proved that a (G) ≤ +2 for almost all -regular graphs.…”

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confidence: 99%