In this note, we prove a structural theorem for planar graphs, namely that every planar graph has one of four possible configurations: (1) a vertex of degree 1, (2) intersecting triangles, (3) an edge xy with d(x)+d(y)≤9, (4) a 2-alternating cycle. Applying this theorem, new moderate results on edge choosability, total choosability, edge-partitions and linear arboricity of planar graphs are obtained. © 2011 Elsevier B.V. All rights reserved.
Sheng, H., & Wang, Y. (2011). A structural theorem for planar graphs with some applications. Discrete Applied Mathematics, 159(11), 1183–1187. https://doi.org/10.1016/j.dam.2011.03.005