Acyclic Edge Coloring of 4-Regular Graphs Without 3-Cycles

Shu, Qiaojun; Wang, Yiqiao; Ma, Yulai; Wang, Weifan

doi:10.1007/s40840-017-0484-x

Cited by 6 publications

(2 citation statements)

References 22 publications

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“…The upper bound was improved to 16∆ [10], to 9.62(∆ − 1) [11], to 4∆ − 4 [6], and most recently in 2017 to 3.74(∆ − 1) + 1 by Giotis et al [8] using the Lovász local lemma. On the other hand, the AECC has been confirmed true for graphs with ∆ ∈ {3, 4} [15,3,4,14,19].…”

Section: Introductionmentioning

confidence: 90%

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Acyclic edge coloring conjecture is true on planar graphs without intersecting triangles

Shu¹,

Lin²,

Miyano³

2020

Preprint

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An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic edge coloring conjecture by Fiamčik (1978) and Alon, Sudakov and Zaks (2001) states that every simple graph with maximum degree ∆ is acyclically edge (∆ + 2)-colorable. Despite many milestones, the conjecture remains open even for planar graphs. In this paper, we confirm affirmatively the conjecture on planar graphs without intersecting triangles. We do so by first showing, by discharging methods, that every planar graph without intersecting triangles must have at least one of the six specified groups of local structures, and then proving the conjecture by recoloring certain edges in each such local structure and by induction on the number of edges in the graph.

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Section: Introductionmentioning

confidence: 90%

“…The following lemma gives the starting point. Lemma 8 ([15,3,4,14,19]) If ∆ ∈ {3, 4}, then a (G) ≤ ∆+2, and an acyclic edge (∆+2)-coloring can be obtained in polynomial time.…”

Section: Acyclic Edge Coloringmentioning

confidence: 99%