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

Cranston, Daniel W.

doi:10.1137/17m1158355

Cited by 5 publications

(1 citation statement)

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“…The acyclic chromatic index has been determined exactly for some classes of graphs like outer planar graphs when ∆ = 4 (Hou and Wu [16], Hou et al [17]), series-parallel graphs when ∆ = 4 (Wang and Shu [23]), planar graphs with girth at least 5 and ∆ ≥ 19 (Basavaraju et al [7]) and planar graphs with ∆ ≥ 4.2 × 10 14 (Cranston [8]). In case of outer planar graphs and series-parallel graphs, a ′ (G) = ∆ if ∆ ≥ 5 and when ∆ = 3, they characterize the graphs that require 4 colors.…”

Section: Introductionmentioning

confidence: 99%

Acyclic Chromatic Index of Chordless Graphs

Basavaraju¹,

Hegde²,

Kulamarva³

2023

Preprint

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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.

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