Optimal acyclic edge‐coloring of cubic graphs

Andersen, Lars Døvling; Máčajová, Edita; Mazák, Ján

doi:10.1002/jgt.20650

Cited by 18 publications

(6 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Otherwise, v is adjacent to at most one vertex whose degree is at most seven, say v 1 . Then v gives 4 5 to each each incident face containing vv 1 and 1 5 to each the other incident face. 4 5 to each incident face.…”

Section: Lemma 21 Let G Be a Connected Planar Graph Then There Exists...mentioning

confidence: 99%

“…Then v gives 4 5 to each each incident face containing vv 1 and 1 5 to each the other incident face. 4 5 to each incident face. (R 3.2 ) If d(v 1 ) ≤ 6 and d(v 2 ) ≥ 8, then v gives 5 4 to each incident face containing vv 1 and 1 2 to each the other incident face.…”

Section: Lemma 21 Let G Be a Connected Planar Graph Then There Exists...mentioning

confidence: 99%

“…Otherwise, d(y) ≤ 7 and d(z) ≥ 10. The vertex x gives 4 5 to f by (R 2 ) and z gives at least 7 5 by (R 1 ). Moreover, y gives at least 5 4 to f by (R 1 ), (R 2 ) and (R 3 ).…”

Section: Lemma 21 Let G Be a Connected Planar Graph Then There Exists...mentioning

confidence: 99%

“…In 2001, Alon, Sudakov and Zaks [2] stated the Acyclic Edge Coloring Conjecture, which says that χ ′ a (G) ≤ ∆(G) + 2 for every graph G. Though the best known upper bound for general case is far from the conjecture ∆(G) + 2, the conjecture has been shown to be true for some special classes of graphs, including graphs with maximum degree at most three [4], non regular graphs with maximum degree at most four [6], outerplanar graphs [11], graphs with large girth [2], and so on. Fiedorowicz et al [9] gave an upper bound of 2∆(G) + 29 for planar graphs and of ∆(G) + 6 for triangle-free planar graphs.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An improved bound on acyclic chromatic index of planar graphs

Guan¹,

Hou²,

Yang³

2012

Preprint

View full text Add to dashboard Cite

A proper edge coloring of a graph G is called acyclic if there is no bichromatic cycle in G. The acyclic chromatic index of G, denoted by χ ′ a (G), is the least number of colors k such that G has an acyclic edge k-coloring. Basavaraju et al. [Acyclic edgecoloring of planar graphs, SIAM J. Discrete Math. 25 (2) (2011), [463][464][465][466][467][468][469][470][471][472][473][474][475][476][477][478] showed that χ ′ a (G) ≤ ∆(G) + 12 for planar graphs G with maximum degree ∆(G). In this paper, the bound is improved to ∆(G) + 10.

show abstract

Section: Lemma 21 Let G Be a Connected Planar Graph Then There Exists...mentioning

confidence: 99%

Section: Lemma 21 Let G Be a Connected Planar Graph Then There Exists...mentioning

confidence: 99%

“…Otherwise, d(y) ≤ 7 and d(z) ≥ 10. The vertex x gives 4 5 to f by (R 2 ) and z gives at least 7 5 by (R 1 ). Moreover, y gives at least 5 4 to f by (R 1 ), (R 2 ) and (R 3 ).…”

Section: Lemma 21 Let G Be a Connected Planar Graph Then There Exists...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An improved bound on acyclic chromatic index of planar graphs

Guan¹,

Hou²,

Yang³

2012

Preprint

View full text Add to dashboard Cite

show abstract

“…We also know that any cubic graph can be colored using at most 5 colors. Andersen et al [5] proved that any cubic graph other than K 4 or K 3,3 can be colored using 4 colors which determines the acyclic chromatic index of cubic graphs exactly. Further, we note that all the above mentioned results are constructive in nature and hence, they also yield a polynomial time algorithm for the optimum coloring.…”

Section: Introductionmentioning

confidence: 99%

Acyclic Chromatic Index of Chordless Graphs

Basavaraju¹,

Hegde²,

Kulamarva³

2023

Preprint

View full text Add to dashboard Cite

An acyclic edge coloring of a graph is a proper edge coloring in which there are no bichromatic cycles. The acyclic chromatic index of a graph G denoted by a ′ (G), is the minimum positive integer k such that G has an acyclic edge coloring with k colors. It has been conjectured by Fiamčík that a ′ (G) ≤ ∆ + 2 for any graph G with maximum degree ∆. Linear arboricity of a graph G, denoted by la(G), is the minimum number of linear forests into which the edges of G can be partitioned. A graph is said to be chordless if no cycle in the graph contains a chord. Every 2-connected chordless graph is a minimally 2-connected graph. It was shown by Basavaraju and Chandran that if G is 2-degenerate, then a ′ (G) ≤ ∆ + 1. Since chordless graphs are also 2-degenerate, we have a ′ (G) ≤ ∆ + 1 for any chordless graph G. Machado, de Figueiredo and Trotignon proved that the chromatic index of a chordless graph is ∆ when ∆ ≥ 3. They also obtained a polynomial time algorithm to color a chordless graph optimally. We improve this result by proving that the acyclic chromatic index of a chordless graph is ∆, except when ∆ = 2 and the graph has a cycle, in which case it is ∆ + 1. We also provide the sketch of a polynomial time algorithm for an optimal acyclic edge coloring of a chordless graph. As a byproduct, we also prove that la(G) = ⌈ ∆ 2 ⌉, unless G has a cycle with ∆ = 2, in which case la(G) = ⌈ ∆+1 2 ⌉ = 2. To obtain the result on acyclic chromatic index, we prove a structural result on chordless graphs which is a refinement of the structure given by Machado, de Figueiredo and Trotignon for this class of graphs. This might be of independent interest.

show abstract