2012
DOI: 10.1002/jgt.21683
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Acyclic Chromatic Indices of Planar Graphs with Girth At Least 4

Abstract: An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a (G) of G is the smallest integer k such that G has an acyclic edge coloring using k colors.

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Cited by 23 publications
(6 citation statements)
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“…Fiamčik [10] and later Alon et al [2] made the following conjecture: Alon et al [1] proved that χ a (G) ≤ 64Δ(G) for any graph G. Molloy and Reed [15] improved this bound to that χ a (G) ≤ 16Δ(G). Něsetřil and Wormald [17] proved that χ a (G) ≤ Δ(G) + 1 for a random Δ(G)-regular graph G. The acyclic edge coloring of some special classes of graphs was also investigated, including subcubic graphs (Basavaraju and Chandran [4]; Fiamčik [10]; Skulrattanakulchai [21]), graphs with maximum degree 4 (Basavaraju and Chandran [3]), outerplanar graphs (Hou et al [13]; Muthu et al [16]), series-parallel graphs (Hou et al [12]; Wang and Shu [22]), and planar graphs (Cohen et al [8]; Dong and Xu [9]; Fiedorowicz et al [11]; Shu and Wang [19,20]; Wang et al [23]; Yu et al [25]). In particular, Conjecture 1.1 was confirmed for planar graphs without 3-cycles by Shu and Wang [19].…”
Section: Introductionmentioning
confidence: 99%
“…Fiamčik [10] and later Alon et al [2] made the following conjecture: Alon et al [1] proved that χ a (G) ≤ 64Δ(G) for any graph G. Molloy and Reed [15] improved this bound to that χ a (G) ≤ 16Δ(G). Něsetřil and Wormald [17] proved that χ a (G) ≤ Δ(G) + 1 for a random Δ(G)-regular graph G. The acyclic edge coloring of some special classes of graphs was also investigated, including subcubic graphs (Basavaraju and Chandran [4]; Fiamčik [10]; Skulrattanakulchai [21]), graphs with maximum degree 4 (Basavaraju and Chandran [3]), outerplanar graphs (Hou et al [13]; Muthu et al [16]), series-parallel graphs (Hou et al [12]; Wang and Shu [22]), and planar graphs (Cohen et al [8]; Dong and Xu [9]; Fiedorowicz et al [11]; Shu and Wang [19,20]; Wang et al [23]; Yu et al [25]). In particular, Conjecture 1.1 was confirmed for planar graphs without 3-cycles by Shu and Wang [19].…”
Section: Introductionmentioning
confidence: 99%
“…Muthu et al [23] proved that χ a (G) ≤ ∆(G) + 1 for every outerplanar graph G. Hou et al [21] proved that χ a (G) = ∆(G) for every outerplanar graph G with ∆(G) ≥ 5. The conjecture is also true for planar graphs with girth at least five [9,20] and planar graphs with girth at least four [26].…”
Section: Introductionmentioning
confidence: 90%
“…We say that two triangles are adjacent if they share a common edge, and are intersecting if they share at least a common vertex. Recall that the truth of the AECC for planar graphs without triangles has been verified in [13]. When a planar graph G contains triangles but no intersecting triangles, Hou, Roussel and Wu [9] proved the upper bound a (G) ≤ ∆ + 5, and Wang and Zhang [17] improved it to a (G) ≤ ∆ + 3.…”
Section: Introductionmentioning
confidence: 95%
“…The upper bound was improved to ∆ + 7 by Wang, Shu and Wang [21] and to ∆ + 6 by Wang and Zhang [18]. The AECC has been confirmed true for planar graphs without i-cycles for each i ∈ {3, 4, 5, 6} in [13,20,12,22], respectively.…”
Section: Introductionmentioning
confidence: 96%