Edge-choosability of planar graphs without adjacent triangles or without 7-cycles

Abstract

A graph G is edge-L-colorable, if for a given edge assignment L = {L (e) : e ∈ E (G)}, there exists a proper edge-coloring φ{symbol} of G such that φ{symbol} (e) ∈ L (e) for all e ∈ E (G). If G is edge-L-colorable for every edge assignment L with | L (e) | ≥ k for e ∈ E (G), then G is said to be edge-k-choosable. In this paper, we prove that if G is a planar graph with maximum degree Δ (G) ≠ 5 and without adjacent 3-cycles, or with maximum degree Δ (G) ≠ 5, 6 and without 7-cycles, then G is edge-(Δ (G) + 1)-choosable. © 2007 Elsevier B.V. All rights reserved.